Question : The given expression is equal to: $(\cot B-\tan B)\sin B\cos B$:
Option 1: $1-2 \sec ^2 B$
Option 2: $1-2 \cos ^2 {B}$
Option 3: $2 \cos ^2 {B}-1$
Option 4: $2 \sec ^2 {B}-1$
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Correct Answer: $2 \cos ^2 {B}-1$
Solution :
$(\cot B-\tan B)\sin B\cos B$
= $(\frac{\cos B}{\sin B}-\frac{\sin B}{\cos B})\sin B\cos B$
= $(\frac{\cos^2 B - \sin^2 B}{\sin B\cos B})\sin B\cos B$
= $\cos^2 B - \sin^2 B$
= $\cos^2 B - (1- \cos^2 B)$
= $2\cos^2 B - 1$
Hence, the correct answer is $2\cos^2 B - 1$.
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