Общий знаменатель (-log(sin(x)+1)/2)+log(sin(x)-1)/2-1/sin(x)

Вы ввели

-log(sin(x) + 1)    log(sin(x) - 1)     1   
----------------- + --------------- - ------
        2                  2          sin(x)

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{1}{2} \left(-1 \log{\left (\sin{\left (x \right )} + 1 \right )}\right) - \frac{1}{\sin{\left (x \right )}}$$

Степени

log(-1 + sin(x))     1      log(1 + sin(x))
---------------- - ------ - ---------------
       2           sin(x)          2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{1}{\sin{\left (x \right )}}$$

Численный ответ

-1/sin(x) + 0.5*log(sin(x) - 1) - 0.5*log(sin(x) + 1)

Рациональный знаменатель

-2 + log(-1 + sin(x))*sin(x) - log(1 + sin(x))*sin(x)
-----------------------------------------------------
                       2*sin(x)                      

$$\frac{1}{2 \sin{\left (x \right )}} \left(\log{\left (\sin{\left (x \right )} - 1 \right )} \sin{\left (x \right )} - \log{\left (\sin{\left (x \right )} + 1 \right )} \sin{\left (x \right )} - 2\right)$$

Объединение рациональных выражений

-2 + (-log(1 + sin(x)) + log(-1 + sin(x)))*sin(x)
-------------------------------------------------
                     2*sin(x)                    

$$\frac{1}{2 \sin{\left (x \right )}} \left(\left(\log{\left (\sin{\left (x \right )} - 1 \right )} - \log{\left (\sin{\left (x \right )} + 1 \right )}\right) \sin{\left (x \right )} - 2\right)$$

Общее упрощение

log(-1 + sin(x))     1      log(1 + sin(x))
---------------- - ------ - ---------------
       2           sin(x)          2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{1}{\sin{\left (x \right )}}$$

Собрать выражение

log(-1 + sin(x))     1      log(1 + sin(x))
---------------- - ------ - ---------------
       2           sin(x)          2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{1}{\sin{\left (x \right )}}$$

log(-1 + sin(x))            log(1 + sin(x))
---------------- - csc(x) - ---------------
       2                           2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \csc{\left (x \right )}$$

Общий знаменатель

log(-1 + sin(x))     1      log(1 + sin(x))
---------------- - ------ - ---------------
       2           sin(x)          2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{1}{\sin{\left (x \right )}}$$

Тригонометрическая часть

-log(1 + sin(x)) + log(-1 + sin(x))     1   
----------------------------------- - ------
                 2                    sin(x)

$$\frac{1}{2} \left(\log{\left (\sin{\left (x \right )} - 1 \right )} - \log{\left (\sin{\left (x \right )} + 1 \right )}\right) - \frac{1}{\sin{\left (x \right )}}$$

Комбинаторика

-(2 + log(1 + sin(x))*sin(x) - log(-1 + sin(x))*sin(x)) 
--------------------------------------------------------
                        2*sin(x)                        

$$- \frac{1}{2 \sin{\left (x \right )}} \left(- \log{\left (\sin{\left (x \right )} - 1 \right )} \sin{\left (x \right )} + \log{\left (\sin{\left (x \right )} + 1 \right )} \sin{\left (x \right )} + 2\right)$$

Раскрыть выражение

log(-1 + sin(x))     1      log(1 + sin(x))
---------------- - ------ - ---------------
       2           sin(x)          2       

$$\frac{1}{2} \log{\left (\sin{\left (x \right )} - 1 \right )} - \frac{1}{2} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{1}{\sin{\left (x \right )}}$$