\(\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=\)

\(\text {Let } I =\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x \)
\( I =\int_{0}^{\frac{\pi}{2}} \frac{1-\frac{\cos x}{\sin x}}{\frac{1}{\sin x}+\cos x} d x=\int_{0}^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\sin x \cos x} d x ...(1)\)
\( I =\int_{0}^{\frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-x\right)}{1+\sin \left(\frac{\pi}{x}-x\right) \cos \left(\frac{\pi}{2}-x\right)} d x \)
\( I =\int_{0}^{\frac{\pi}{2}} \frac{\cos x-\sin x}{1+\cos x \sin x} d x ...(2)\)
Adding equations (1) \& (2), we get
\(\begin{array}{l}
2 I=\int_{0}^{\frac{\pi}{2}} \frac{\sin x-\cos x+\cos x-\sin x}{1+\sin x \cos x} d x \\
2 I=0 \Rightarrow I=0
\end{array}\)