Question

Prove that :

\[ \frac{\tan\left(\frac{\pi}{2}-x\right)\sec(\pi-x)\sin(-x)} {\sin(\pi+x)\cot(2\pi-x)\cosec\left(\frac{\pi}{2}-x\right)} =1 \]

Solution

\[ \begin{aligned} &\frac{\tan\left(\frac{\pi}{2}-x\right)\sec(\pi-x)\sin(-x)} {\sin(\pi+x)\cot(2\pi-x)\cosec\left(\frac{\pi}{2}-x\right)} \\[8pt] =& \frac{\cot x(-\sec x)(-\sin x)} {(-\sin x)(\cot x)(\sec x)} \\[8pt] =& \frac{\cot x\sec x\sin x} {\sin x\cot x\sec x} \\[8pt] =& 1 \end{aligned} \]

Hence Proved.