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Другие калькуляторы

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Общий знаменатель:
cos(a)/1-cos(a)-sin(a)/cos(a)+1
x^2/x^2+7*x*y/x/x^2-49*y^2
(1/sqrt(1+x)+1/sqrt(1-x))*(sqrt(1+x)+sqrt(1-x))
b+2/b^2-3*b+9-6/b^3+27
cos(3*a-2*b)*cos(a-2*b)+sin(3*a-2*b)*sin(a-2*b) если a=3/2
18*b*4 если b=4
sin(-x) если x=4
26+m+34 если m=3
Разложить многочлен на множители:
3*x^2-12*x+9
x^3+4*x^2-x-4
4*p-24*p*g+36*p*g^2
n^3+15*n^2-75*n+125
Умножение столбиком:
45*560
35*3576
148*226
40*1480
Деление столбиком:
862/6
942/6
1518/6
1737/6
Раскрыть скобки в:
(x+y)*(x-y)-(x^2-3*y^2)
(f3+f4+f5+f6)*(f1+f2)*(f1+f2+f5+f6)*(f2+f4+f6)*(f3+f4+f7)*f7*(f1+f3+f5)
(a+b-c)*(a^2-b^2)
6*x*(5*x-24)-4*(3-2*x)^2
Разложение числа на простые множители:
9287
8254
1170
442
Идентичные выражения
cos(a)/ один -cos(a)-sin(a)/cos(a)+ один
косинус от (a) делить на 1 минус косинус от (a) минус синус от (a) делить на косинус от (a) плюс 1
косинус от (a) делить на один минус косинус от (a) минус синус от (a) делить на косинус от (a) плюс один
cosa/1-cosa-sina/cosa+1
cos(a) разделить на 1-cos(a)-sin(a) разделить на cos(a)+1
Похожие выражения
cos(a)/1-cos(a)+sin(a)/cos(a)+1
cos(a)/1-cos(a)-sin(a)/cos(a)-1
cos(a)/1+cos(a)-sin(a)/cos(a)+1

Упрощение выражений/ Общий знаменатель/ cos(a)/1-cos(a)-sin(a)/cos(a)+1

Общий знаменатель cos(a)/1-cos(a)-sin(a)/cos(a)+1

Решение

Вы ввели [src]

cos(a)            sin(a)    
------ - cos(a) - ------ + 1
  1               cos(a)

$$- \cos{\left(a \right)} + \frac{\cos{\left(a \right)}}{1} - \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

cos(a)/1 - cos(a) - sin(a)/cos(a) + 1

Общее упрощение [src]

1 - tan(a)

$$- \tan{\left(a \right)} + 1$$

1 - tan(a)

Общий знаменатель [src]

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

1 - sin(a)/cos(a)

Комбинаторика [src]

-sin(a) + cos(a)
----------------
     cos(a)

$$\frac{- \sin{\left(a \right)} + \cos{\left(a \right)}}{\cos{\left(a \right)}}$$

(-sin(a) + cos(a))/cos(a)

Численный ответ [src]

1.0 - sin(a)/cos(a)

1.0 - sin(a)/cos(a)

Объединение рациональных выражений [src]

-sin(a) + cos(a)
----------------
     cos(a)

$$\frac{- \sin{\left(a \right)} + \cos{\left(a \right)}}{\cos{\left(a \right)}}$$

(-sin(a) + cos(a))/cos(a)

Рациональный знаменатель [src]

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

-sin(a) + cos(a)
----------------
     cos(a)

$$\frac{- \sin{\left(a \right)} + \cos{\left(a \right)}}{\cos{\left(a \right)}}$$

(-sin(a) + cos(a))/cos(a)

Степени [src]

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

      /   -I*a    I*a\
    I*\- e     + e   /
1 + ------------------
       / I*a    -I*a\ 
       |e      e    | 
     2*|---- + -----| 
       \ 2       2  /

$$1 + \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}$$

1 + i*(-exp(-i*a) + exp(i*a))/(2*(exp(i*a)/2 + exp(-i*a)/2))

Раскрыть выражение [src]

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

1 - sin(a)/cos(a)

Собрать выражение [src]

1 - tan(a)

$$- \tan{\left(a \right)} + 1$$

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

1 - sin(a)/cos(a)

Тригонометрическая часть [src]

1 - tan(a)

$$- \tan{\left(a \right)} + 1$$

      1   
1 - ------
    cot(a)

$$1 - \frac{1}{\cot{\left(a \right)}}$$

    sec(a)
1 - ------
    csc(a)

$$1 - \frac{\sec{\left(a \right)}}{\csc{\left(a \right)}}$$

    sin(a)
1 - ------
    cos(a)

$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1$$

         2   
    2*sin (a)
1 - ---------
     sin(2*a)

$$- \frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1$$

       /    pi\
    cos|a - --|
       \    2 /
1 - -----------
       cos(a)

$$1 - \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$

       /pi    \
    csc|-- - a|
       \2     /
1 - -----------
       csc(a)

$$1 - \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$

       sec(a)  
1 - -----------
       /    pi\
    sec|a - --|
       \    2 /

$$- \frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1$$

       sin(a)  
1 - -----------
       /    pi\
    sin|a + --|
       \    2 /

$$- \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)}} + 1$$

       sec(a)  
1 - -----------
       /pi    \
    sec|-- - a|
       \2     /

$$- \frac{\sec{\left(a \right)}}{\sec{\left(- a + \frac{\pi}{2} \right)}} + 1$$

       /pi    \
    csc|-- - a|
       \2     /
1 - -----------
    csc(pi - a)

$$1 - \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(- a + \pi \right)}}$$

    /      1   \    /a\
1 - |1 + ------|*tan|-|
    \    cos(a)/    \2/

$$- \left(1 + \frac{1}{\cos{\left(a \right)}}\right) \tan{\left(\frac{a}{2} \right)} + 1$$

           /a\ 
      2*tan|-| 
           \2/ 
1 - -----------
           2/a\
    1 - tan |-|
            \2/

$$1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}$$

             2          
1 - --------------------
    /       1   \    /a\
    |1 - -------|*cot|-|
    |       2/a\|    \2/
    |    cot |-||       
    \        \2//

$$1 - \frac{2}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}$$

           2/a\        
      4*sin |-|*sin(a) 
            \2/        
1 - -------------------
       2           4/a\
    sin (a) - 4*sin |-|
                    \2/

$$- \frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + 1$$

               2/a\       
          4*sin |-|       
                \2/       
1 - ----------------------
    /         4/a\\       
    |    4*sin |-||       
    |          \2/|       
    |1 - ---------|*sin(a)
    |        2    |       
    \     sin (a) /

$$1 - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}$$

               /a   pi\      
          2*cos|- - --|      
               \2   2 /      
1 - -------------------------
    /       2/a   pi\\       
    |    cos |- - --||       
    |        \2   2 /|    /a\
    |1 - ------------|*cos|-|
    |         2/a\   |    \2/
    |      cos |-|   |       
    \          \2/   /

$$1 - \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}$$

                    /a\           
               2*sec|-|           
                    \2/           
1 - ------------------------------
    /         2/a\   \            
    |      sec |-|   |            
    |          \2/   |    /a   pi\
    |1 - ------------|*sec|- - --|
    |       2/a   pi\|    \2   2 /
    |    sec |- - --||            
    \        \2   2 //

$$1 - \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$

               /pi   a\      
          2*csc|-- - -|      
               \2    2/      
1 - -------------------------
    /       2/pi   a\\       
    |    csc |-- - -||       
    |        \2    2/|    /a\
    |1 - ------------|*csc|-|
    |         2/a\   |    \2/
    |      csc |-|   |       
    \          \2/   /

$$1 - \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}$$

    /       2/a   pi\\    /a\
    |1 + tan |- + --||*tan|-|
    \        \2   4 //    \2/
1 - -------------------------
    /       2/a\\    /a   pi\
    |1 + tan |-||*tan|- + --|
    \        \2//    \2   4 /

$$1 - \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$

    /       2/a   pi\\    /a\
    |1 + tan |- + --||*cot|-|
    \        \2   4 //    \2/
1 - -------------------------
    /       2/a\\    /a   pi\
    |1 + cot |-||*tan|- + --|
    \        \2//    \2   4 /

$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1$$

    /       2/a   pi\\             
    |1 - cot |- + --||*(1 + sin(a))
    \        \2   4 //             
1 - -------------------------------
          /       2/a\\    2/a\    
        2*|1 - tan |-||*cos |-|    
          \        \2//     \2/

$$1 - \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}$$

    //  0     for a mod pi = 0\ //  1     for a mod 2*pi = 0\
1 - |<                        |*|<                          |
    \\sin(a)     otherwise    / \\sec(a)      otherwise     /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sec{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$

    /       2/a\\ /        2/a   pi\\
    |1 + cot |-||*|-1 + tan |- + --||
    \        \2// \         \2   4 //
1 - ---------------------------------
    /       2/a   pi\\ /        2/a\\
    |1 + tan |- + --||*|-1 + cot |-||
    \        \2   4 // \         \2//

$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)} + 1$$

    /       2/a\\ /       2/a   pi\\
    |1 + tan |-||*|1 - cot |- + --||
    \        \2// \        \2   4 //
1 - --------------------------------
    /       2/a   pi\\ /       2/a\\
    |1 + cot |- + --||*|1 - tan |-||
    \        \2   4 // \        \2//

$$1 - \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$

                                //  1     for a mod 2*pi = 0\
    //  0     for a mod pi = 0\ ||                          |
1 - |<                        |*|<  1                       |
    \\sin(a)     otherwise    / ||------      otherwise     |
                                \\cos(a)                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //     0       for a mod pi = 0\ //  1     for a mod 2*pi = 0\
    ||                             | ||                          |
1 - |<   /    pi\                  |*|<  1                       |
    ||cos|a - --|     otherwise    | ||------      otherwise     |
    \\   \    2 /                  / \\cos(a)                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                //     1       for a mod 2*pi = 0\
                                ||                               |
    //  0     for a mod pi = 0\ ||     1                         |
1 - |<                        |*|<-----------      otherwise     |
    \\sin(a)     otherwise    / ||   /    pi\                    |
                                ||sin|a + --|                    |
                                \\   \    2 /                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sin{\left(a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //     0       for a mod pi = 0\                              
    ||                             |                              
    ||     1                       | //  1     for a mod 2*pi = 0\
1 - |<-----------     otherwise    |*|<                          |
    ||   /    pi\                  | \\sec(a)      otherwise     /
    ||sec|a - --|                  |                              
    \\   \    2 /                  /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sec{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //  0     for a mod pi = 0\ //     1       for a mod 2*pi = 0\
    ||                        | ||                               |
1 - |<  1                     |*|<   /pi    \                    |
    ||------     otherwise    | ||csc|-- - a|      otherwise     |
    \\csc(a)                  / \\   \2     /                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\csc{\left(- a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //  1     for a mod 2*pi = 0\ //            /    3*pi\             \
    ||                          | ||  1     for |a + ----| mod 2*pi = 0|
1 - |<  1                       |*|<            \     2  /             |
    ||------      otherwise     | ||                                   |
    \\cos(a)                    / \\sin(a)           otherwise         /

$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //    0       for a mod pi = 0\                              
    ||                            | //  1     for a mod 2*pi = 0\
    ||1 - cos(a)                  | ||                          |
1 - |<----------     otherwise    |*|<  1                       |
    ||     /a\                    | ||------      otherwise     |
    ||  tan|-|                    | \\cos(a)                    /
    \\     \2/                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //     0       for a mod pi = 0\ //     1        for a mod 2*pi = 0\
    ||                             | ||                                |
    ||       /a\                   | ||       2/a\                     |
    ||  2*cot|-|                   | ||1 + cot |-|                     |
1 - |<       \2/                   |*|<        \2/                     |
    ||-----------     otherwise    | ||------------      otherwise     |
    ||       2/a\                  | ||        2/a\                    |
    ||1 + cot |-|                  | ||-1 + cot |-|                    |
    \\        \2/                  / \\         \2/                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                //                    /    pi\           \
                                ||     zoo        for |a + --| mod pi = 0|
                                ||                    \    2 /           |
                                ||                                       |
    //  0     for a mod pi = 0\ ||    /a   pi\                           |
1 - |<                        |*|< tan|- + --|                           |
    \\sin(a)     otherwise    / ||    \2   4 /                           |
                                ||--------------         otherwise       |
                                ||     2/a   pi\                         |
                                ||2*sin |- + --|                         |
                                \\      \2   4 /                         /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //     0       for a mod pi = 0\ //     1       for a mod 2*pi = 0\
    ||                             | ||                               |
    ||       /a\                   | ||       2/a\                    |
    ||  2*tan|-|                   | ||1 + tan |-|                    |
1 - |<       \2/                   |*|<        \2/                    |
    ||-----------     otherwise    | ||-----------      otherwise     |
    ||       2/a\                  | ||       2/a\                    |
    ||1 + tan |-|                  | ||1 - tan |-|                    |
    \\        \2/                  / \\        \2/                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                              //     1        for a mod 2*pi = 0\
                                              ||                                |
    //         0            for a mod pi = 0\ ||       1                        |
    ||                                      | ||1 + -------                     |
    ||         2                            | ||       2/a\                     |
    ||--------------------     otherwise    | ||    tan |-|                     |
1 - |

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                                   //             1               for a mod 2*pi = 0\
    //            0              for a mod pi = 0\ ||                                               |
    ||                                           | ||/  1     for a mod 2*pi = 0                    |
1 - |

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                     //                      /    pi\           \
    //     0       for a mod pi = 0\ ||      zoo         for |a + --| mod pi = 0|
    ||                             | ||                      \    2 /           |
    ||       /a\                   | ||                                         |
    ||  2*cot|-|                   | ||       2/a   pi\                         |
1 - |<       \2/                   |*|<1 + cot |- + --|                         |
    ||-----------     otherwise    | ||        \2   4 /                         |
    ||       2/a\                  | ||----------------         otherwise       |
    ||1 + cot |-|                  | ||      /a   pi\                           |
    \\        \2/                  / || 2*cot|- + --|                           |
                                     \\      \2   4 /                           /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                        //                       /    3*pi\             \
    //     1        for a mod 2*pi = 0\ ||        1          for |a + ----| mod 2*pi = 0|
    ||                                | ||                       \     2  /             |
    ||       2/a\                     | ||                                              |
    ||1 + cot |-|                     | ||        2/a   pi\                             |
1 - |<        \2/                     |*|<-1 + tan |- + --|                             |
    ||------------      otherwise     | ||         \2   4 /                             |
    ||        2/a\                    | ||-----------------           otherwise         |
    ||-1 + cot |-|                    | ||        2/a   pi\                             |
    \\         \2/                    / || 1 + tan |- + --|                             |
                                        \\         \2   4 /                             /

$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //             0                for a mod pi = 0\                                                 
    ||                                              |                                                 
    ||          2*sin(a)                            | //            1              for a mod 2*pi = 0\
    ||----------------------------     otherwise    | ||                                             |
    ||             /        2    \                  | ||             2      2                        |
1 - |<             |     sin (a) |                  |*|< (1 - cos(a))  + sin (a)                     |
    ||(1 - cos(a))*|1 + ---------|                  | ||-------------------------      otherwise     |
    ||             |         4/a\|                  | ||          2                                  |
    ||             |    4*sin |-||                  | \\-2 + 2*sin (a) + 2*cos(a)                    /
    ||             \          \2//                  |                                                 
    \\                                              /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \cos{\left(a \right)} + 1\right)^{2} + \sin^{2}{\left(a \right)}}{2 \sin^{2}{\left(a \right)} + 2 \cos{\left(a \right)} - 2} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                                 //      1         for a mod 2*pi = 0\
                                                 ||                                  |
    //           0             for a mod pi = 0\ ||        2                         |
    ||                                         | ||     sin (a)                      |
    ||         sin(a)                          | ||1 + ---------                     |
    ||-----------------------     otherwise    | ||         4/a\                     |
    ||/        2    \                          | ||    4*sin |-|                     |
1 - |<|     sin (a) |    2/a\                  |*|<          \2/                     |
    |||1 + ---------|*sin |-|                  | ||--------------      otherwise     |
    |||         4/a\|     \2/                  | ||         2                        |
    |||    4*sin |-||                          | ||      sin (a)                     |
    ||\          \2//                          | ||-1 + ---------                    |
    \\                                         / ||          4/a\                    |
                                                 ||     4*sin |-|                    |
                                                 \\           \2/                    /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$

    //              0                 for a mod pi = 0\ //                1                  for a mod 2*pi = 0\
    ||                                                | ||                                                     |
    ||/     0       for a mod pi = 0                  | ||/     1        for a mod 2*pi = 0                    |
    |||                                               | |||                                                    |
    |||       /a\                                     | |||       2/a\                                         |
1 - |<|  2*cot|-|                                     |*|<|1 + cot |-|                                         |
    ||<       \2/                        otherwise    | ||<        \2/                           otherwise     |
    |||-----------     otherwise                      | |||------------      otherwise                         |
    |||       2/a\                                    | |||        2/a\                                        |
    |||1 + cot |-|                                    | |||-1 + cot |-|                                        |
    \\\        \2/                                    / \\\         \2/                                        /

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                                        //        1          for a mod 2*pi = 0\
                                                        ||                                     |
    //              0                 for a mod pi = 0\ ||          2/a\                       |
    ||                                                | ||       cos |-|                       |
    ||                /a\                             | ||           \2/                       |
    ||           2*cos|-|                             | || 1 + ------------                    |
    ||                \2/                             | ||        2/a   pi\                    |
    ||------------------------------     otherwise    | ||     cos |- - --|                    |
1 - |

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                                   //        1          for a mod 2*pi = 0\
                                                   ||                                     |
    //            0              for a mod pi = 0\ ||        2/a   pi\                    |
    ||                                           | ||     sec |- - --|                    |
    ||           /a   pi\                        | ||         \2   2 /                    |
    ||      2*sec|- - --|                        | || 1 + ------------                    |
    ||           \2   2 /                        | ||          2/a\                       |
    ||-------------------------     otherwise    | ||       sec |-|                       |
1 - |

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$

                                                        //        1          for a mod 2*pi = 0\
                                                        ||                                     |
    //              0                 for a mod pi = 0\ ||          2/a\                       |
    ||                                                | ||       csc |-|                       |
    ||                /a\                             | ||           \2/                       |
    ||           2*csc|-|                             | || 1 + ------------                    |
    ||                \2/                             | ||        2/pi   a\                    |
    ||------------------------------     otherwise    | ||     csc |-- - -|                    |
1 - |

$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right)\right) + 1$$

1 - Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True))*Piecewise((1, Mod(a = 2*pi, 0)), ((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(-1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))

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