Question

Prove that :

\[ \frac{\cosec(90^\circ+x)+\cot(450^\circ+x)} {\cosec(90^\circ-x)+\tan(180^\circ-x)} + \frac{\tan(180^\circ+x)\sec(180^\circ-x)} {\tan(360^\circ+x)-\sec(-x)} =2 \]

Solution

\[ \begin{aligned} &\frac{\cosec(90^\circ+x)+\cot(450^\circ+x)} {\cosec(90^\circ-x)+\tan(180^\circ-x)} + \frac{\tan(180^\circ+x)\sec(180^\circ-x)} {\tan(360^\circ+x)-\sec(-x)} \\[8pt] =& \frac{\sec x-\tan x} {\sec x-\tan x} + \frac{\tan x(-\sec x)} {\tan x-\sec x} \\[8pt] =& 1+ \frac{\tan x\sec x} {\sec x-\tan x} \\[8pt] =& 1+1 \\[8pt] =& 2 \end{aligned} \]

Hence Proved.