Question

Prove that :

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

Solution

Using standard identities,

\[ \cos(2\pi+x)=\cos x \]

\[ \cosec(2\pi+x)=\cosec x \]

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

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

\[ \cot(\pi+x)=\cot x \]

Substituting these values,

\[ \begin{aligned} &\frac{\cos x\cdot\cosec x\cdot(-\cot x)} {(-\cosec x)\cos x\cot x} \\[4pt] =& \frac{-\cos x\cosec x\cot x} {-\cos x\cosec x\cot x} \\[4pt] =& 1 \end{aligned} \]

Hence Proved.