sin(t)^2-1/(cos(t)^2)-1+tan(t)*cot(t) если t=1/4 (упростите выражение)

Вы ввели

   2         1                       
sin (t) - ------- - 1 + tan(t)*cot(t)
             2                       
          cos (t)                    

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

Подстановка условия

sin(t)^2 - 1/(cos(t)^2) - 1 + tan(t)*cot(t) при t = 1/4

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

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

sin((1/4))^2 - 1/(cos((1/4))^2) - 1 + tan((1/4))*cot((1/4))

$$\sin^{2}{\left ((1/4) \right )} + -1 \left(\cos^{2}{\left ((1/4) \right )}\right)^{-1} + -1 + \tan{\left ((1/4) \right )} \cot{\left ((1/4) \right )}$$

sin(1/4)^2 - 1/(cos(1/4)^2) - 1 + tan(1/4)*cot(1/4)

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

-1 + sin(1/4)^2 - 1/cos(1/4)^2 + cot(1/4)*tan(1/4)

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

Степени

        2         1                   
-1 + sin (t) - ------- + cot(t)*tan(t)
                  2                   
               cos (t)                

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

Численный ответ

-1.0 + sin(t)^2 - 1/cos(t)^2 + cot(t)*tan(t)

Рациональный знаменатель

        2         2       2         2                 
-1 - cos (t) + cos (t)*sin (t) + cos (t)*cot(t)*tan(t)
------------------------------------------------------
                          2                           
                       cos (t)                        

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

Объединение рациональных выражений

        2         2       2         2                 
-1 - cos (t) + cos (t)*sin (t) + cos (t)*cot(t)*tan(t)
------------------------------------------------------
                          2                           
                       cos (t)                        

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

Общее упрощение

        4   
     sin (t)
-1 - -------
        2   
     cos (t)

$$- \frac{\sin^{4}{\left (t \right )}}{\cos^{2}{\left (t \right )}} - 1$$

Собрать выражение

        2         1                   
-1 + sin (t) - ------- + tan(t)*cot(t)
                  2                   
               cos (t)                

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

1      2      cos(2*t)
- - sec (t) - --------
2                2    

$$- \frac{1}{2} \cos{\left (2 t \right )} - \sec^{2}{\left (t \right )} + \frac{1}{2}$$

Общий знаменатель

        2         1                   
-1 + sin (t) - ------- + cot(t)*tan(t)
                  2                   
               cos (t)                

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

Тригонометрическая часть

        4   
     sin (t)
-1 - -------
        2   
     cos (t)

$$- \frac{\sin^{4}{\left (t \right )}}{\cos^{2}{\left (t \right )}} - 1$$

Комбинаторика

        2         2       2         2                 
-1 - cos (t) + cos (t)*sin (t) + cos (t)*cot(t)*tan(t)
------------------------------------------------------
                          2                           
                       cos (t)                        

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

Раскрыть выражение

        2         1                   
-1 + sin (t) - ------- + cot(t)*tan(t)
                  2                   
               cos (t)                

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