Question

Prove that :

\[ \sec\left(\frac{3\pi}{2}-x\right)\sec\left(x-\frac{5\pi}{2}\right) + \tan\left(\frac{5\pi}{2}+x\right)\tan\left(x-\frac{3\pi}{2}\right) =-1 \]

Solution

\[ \begin{aligned} &\sec\left(\frac{3\pi}{2}-x\right)\sec\left(x-\frac{5\pi}{2}\right) + \tan\left(\frac{5\pi}{2}+x\right)\tan\left(x-\frac{3\pi}{2}\right) \\[8pt] =& (-\cosec x)(\cosec x) + (-\cot x)(\cot x) \\[8pt] =& -\cosec^2x-\cot^2x \\[8pt] =& -(1+\cot^2x)-\cot^2x \\[8pt] =& -1 \end{aligned} \]

Hence Proved.