Evaluate: `inttan^(-1){sqrt(((1-sinx)/(1+sinx)))}\ dx ,\ -pi//2

Explanation

Solution:

According to question, `-\frac{\pi x}{4}-\frac{x^{2}}{4}+C`

I =`\int \tan ^{-1} \sqrt{\frac{1-\cos (\frac{\pi}{2}-x)}{1+\cos (\frac{\pi}{2}-x)}} d x `

=`\int \tan ^{-1} \sqrt{\frac{2 \sin ^{2}(\frac{\pi}{4}-\frac{x}{2})}{2 \cos ^{2}(\frac{\pi}{4}-\frac{x}{2})}} d x=\int \tan ^{-1}\{\tan (\frac{\pi}{4}-\frac{x}{2})\} d x`

=`\int(\frac{\pi}{4}-\frac{x}{2}) d x=\frac{\pi x}{4}-\frac{x^{2}}{4}+C .`