(1+cos(x)/(2*sqrt(sin(x)) ... (x+sqrt(sin(x))) если x=1 (упростите выражение)

Вы ввели

/       cos(x)   \ /        2/      ________\\
|1 + ------------|*\-1 - cot \x + \/ sin(x) //
|        ________|                            
\    2*\/ sin(x) /                            
----------------------------------------------
                /      ________\              
             cot\x + \/ sin(x) /              

$$\frac{1}{\cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}} \left(1 + \frac{\cos{\left (x \right )}}{2 \sqrt{\sin{\left (x \right )}}}\right) \left(- \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} - 1\right)$$

Подстановка условия

((1 + cos(x)*1/(2*sqrt(sin(x))))*(-1 - cot(x + sqrt(sin(x)))^2))*1/cot(x + sqrt(sin(x))) при x = 1

((1 + cos(x)*1/(2*sqrt(sin(x))))*(-1 - cot(x + sqrt(sin(x)))^2))*1/cot(x + sqrt(sin(x)))

$$\left(1 + 1 \frac{2}{\sqrt{\sin{\left (x \right )}}} \cos{\left (x \right )}\right) \left(-1 \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} + -1\right) \cot^{-1}{\left (x + \sqrt{\sin{\left (x \right )}} \right )}$$

((1 + cos((1))*1/(2*sqrt(sin((1)))))*(-1 - cot((1) + sqrt(sin((1))))^2))*1/cot((1) + sqrt(sin((1))))

$$\left(1 + 1 \frac{2}{\sqrt{\sin{\left ((1) \right )}}} \cos{\left ((1) \right )}\right) \left(-1 \cot^{2}{\left ((1) + \sqrt{\sin{\left ((1) \right )}} \right )} + -1\right) \cot^{-1}{\left ((1) + \sqrt{\sin{\left ((1) \right )}} \right )}$$

((1 + cos(1)*1/(2*sqrt(sin(1))))*(-1 - cot(1 + sqrt(sin(1)))^2))*1/cot(1 + sqrt(sin(1)))

$$\left(-1 + -1 \cot^{2}{\left (\sqrt{\sin{\left (1 \right )}} + 1 \right )}\right) \left(1 \frac{2}{\sqrt{\sin{\left (1 \right )}}} \cos{\left (1 \right )} + 1\right) \cot^{-1}{\left (\sqrt{\sin{\left (1 \right )}} + 1 \right )}$$

1/cot(1 + sqrt(sin(1)))*(1 + 2*cos(1)/sqrt(sin(1)))*(-1 - cot(1 + sqrt(sin(1)))^2)

$$\left(-1 + -1 \cot^{2}{\left (\sqrt{\sin{\left (1 \right )}} + 1 \right )}\right) \left(1 + \frac{2 \cos{\left (1 \right )}}{\sqrt{\sin{\left (1 \right )}}}\right) \cot^{-1}{\left (\sqrt{\sin{\left (1 \right )}} + 1 \right )}$$

Численный ответ

(1.0 + 0.5*sin(x)^(-0.5)*cos(x))*(-1.0 - cot(x + sqrt(sin(x)))^2)/cot(x + sqrt(sin(x)))

Рациональный знаменатель

              ________      2/      ________\               2/      ________\   ________
-cos(x) - 2*\/ sin(x)  - cot \x + \/ sin(x) /*cos(x) - 2*cot \x + \/ sin(x) /*\/ sin(x) 
----------------------------------------------------------------------------------------
                                 /      ________\   ________                            
                            2*cot\x + \/ sin(x) /*\/ sin(x)                             

$$\frac{1}{2 \sqrt{\sin{\left (x \right )}} \cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}} \left(- 2 \sqrt{\sin{\left (x \right )}} \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} - 2 \sqrt{\sin{\left (x \right )}} - \cos{\left (x \right )} \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} - \cos{\left (x \right )}\right)$$

Объединение рациональных выражений

/        2/      ________\\ /    ________         \
\-1 - cot \x + \/ sin(x) //*\2*\/ sin(x)  + cos(x)/
---------------------------------------------------
               /      ________\   ________         
          2*cot\x + \/ sin(x) /*\/ sin(x)          

$$\frac{\left(2 \sqrt{\sin{\left (x \right )}} + \cos{\left (x \right )}\right) \left(- \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} - 1\right)}{2 \sqrt{\sin{\left (x \right )}} \cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}}$$

Общее упрощение

              /    ________         \               
             -\2*\/ sin(x)  + cos(x)/               
----------------------------------------------------
     /      ________\   ________    /      ________\
2*cos\x + \/ sin(x) /*\/ sin(x) *sin\x + \/ sin(x) /

$$- \frac{2 \sqrt{\sin{\left (x \right )}} + \cos{\left (x \right )}}{2 \sqrt{\sin{\left (x \right )}} \sin{\left (x + \sqrt{\sin{\left (x \right )}} \right )} \cos{\left (x + \sqrt{\sin{\left (x \right )}} \right )}}$$

Собрать выражение

/                           ________   ________    2/      ________\\                    
|     2/      ________\   \/ cos(x) *\/ cot(x) *csc \x + \/ sin(x) /|    /      ________\
|- csc \x + \/ sin(x) / - ------------------------------------------|*tan\x + \/ sin(x) /
\                                             2                     /                    

$$\left(- \frac{1}{2} \sqrt{\cos{\left (x \right )}} \sqrt{\cot{\left (x \right )}} \csc^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} - \csc^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )}\right) \tan{\left (x + \sqrt{\sin{\left (x \right )}} \right )}$$

Комбинаторика

 /       2/      ________\\ /    ________         \ 
-\1 + cot \x + \/ sin(x) //*\2*\/ sin(x)  + cos(x)/ 
----------------------------------------------------
               /      ________\   ________          
          2*cot\x + \/ sin(x) /*\/ sin(x)           

$$- \frac{\left(2 \sqrt{\sin{\left (x \right )}} + \cos{\left (x \right )}\right) \left(\cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} + 1\right)}{2 \sqrt{\sin{\left (x \right )}} \cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}}$$

Общий знаменатель

                            ________      2/      ________\                
     /      ________\   2*\/ sin(x)  + cot \x + \/ sin(x) /*cos(x) + cos(x)
- cot\x + \/ sin(x) / - ---------------------------------------------------
                                       /      ________\   ________         
                                  2*cot\x + \/ sin(x) /*\/ sin(x)          

$$- \frac{2 \sqrt{\sin{\left (x \right )}} + \cos{\left (x \right )} \cot^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )} + \cos{\left (x \right )}}{2 \sqrt{\sin{\left (x \right )}} \cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}} - \cot{\left (x + \sqrt{\sin{\left (x \right )}} \right )}$$

Тригонометрическая часть

 /       cos(x)   \    /      ________\ 
-|1 + ------------|*tan\x + \/ sin(x) / 
 |        ________|                     
 \    2*\/ sin(x) /                     
----------------------------------------
             2/      ________\          
          sin \x + \/ sin(x) /          

$$- \frac{\tan{\left (x + \sqrt{\sin{\left (x \right )}} \right )}}{\sin^{2}{\left (x + \sqrt{\sin{\left (x \right )}} \right )}} \left(1 + \frac{\cos{\left (x \right )}}{2 \sqrt{\sin{\left (x \right )}}}\right)$$

Раскрыть выражение

                   /                                  2\                           
                   |     /               /  ________\\ |                           
/       cos(x)   \ |     \-1 + cot(x)*cot\\/ sin(x) // | /            /  ________\\
|1 + ------------|*|-1 - ------------------------------|*\cot(x) + cot\\/ sin(x) //
|        ________| |                                2  |                           
\    2*\/ sin(x) / |      /            /  ________\\   |                           
                   \      \cot(x) + cot\\/ sin(x) //   /                           
-----------------------------------------------------------------------------------
                                           /  ________\                            
                            -1 + cot(x)*cot\\/ sin(x) /                            

$$\frac{1}{\cot{\left (x \right )} \cot{\left (\sqrt{\sin{\left (x \right )}} \right )} - 1} \left(1 + \frac{\cos{\left (x \right )}}{2 \sqrt{\sin{\left (x \right )}}}\right) \left(- \frac{\left(\cot{\left (x \right )} \cot{\left (\sqrt{\sin{\left (x \right )}} \right )} - 1\right)^{2}}{\left(\cot{\left (x \right )} + \cot{\left (\sqrt{\sin{\left (x \right )}} \right )}\right)^{2}} - 1\right) \left(\cot{\left (x \right )} + \cot{\left (\sqrt{\sin{\left (x \right )}} \right )}\right)$$