Question : What will be the value of $\cos x \operatorname{cosec} x - \sin x \sec x$?
Option 1: $\cot \frac{2x}{2}$
Option 2: $\tan 2 x $
Option 3: $\cot 2 x $
Option 4: $ 2 \cot 2x$
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Correct Answer: $ 2 \cot 2x$
Solution :
Given: $\cos x \operatorname{cosec} x – \sin x \sec x$
= $\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}$
= $\frac{\cos x \cos x - \sin x \sin x}{\cos x \sin x}$
= $\frac{2(\cos^2 x - \sin^2 x)}{2\sin x \cos x}$
= $\frac{2\cos 2x}{\sin 2x}$
= $2 \cot 2x$
Hence, the correct answer is $2\cot 2x$.
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