Question

Prove that :

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

Solution

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

Hence Proved.