Question

Solve the following problems.

$\frac{d y}{d t} \tan y=\sin (t+y)+\sin (t+y)$

 

Answer 1

Solution 

Solve the following problems.

         $\frac{d y}{d t} \tan y=\sin (t+y)+\sin (t+y)$

$\Longrightarrow \frac{d y}{d t} \tan y=2 \sin t \cos y$

$\Longrightarrow \frac{\tan y}{\cos y} d y=2 \sin t d t$

$\Longrightarrow \int \frac{\tan y}{\cos y} d y=2 \int \sin t d t$

$\Longrightarrow \sec y=-2 \cos t+C$

Above integral explains below:

$\int \frac{\tan y}{\cos y} d y =\int \tan y \cdot \frac{1}{\cos y} d y$

                  $=\int \tan y \sec y d y=\sec y+C$

 

Therefore.   $\sec y=-2 \cos t+C$