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  4. If A, B, C are angles of a triangle, then 2 sin (A/2) textcosec (B/2) sin (C/2)- sin A cot (B/2)- cos A is

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Q. If $A, B, C$ are angles of a triangle, then $2 \sin \frac{A}{2} \text{cosec} \frac{B}{2}\sin \frac{C}{2}-\sin A \cot \frac{B}{2}-\cos A$ is

Trigonometric Functions

Solution:

$2 \sin \frac{A}{2} \operatorname{cosec} \frac{B}{2}\left(\sin \frac{C}{2}-\cos \frac{A}{2} \cos \frac{B}{2}\right)-\cos A$
$=2 \sin \frac{A}{2} \text{cosec} \frac{B}{2}\left(\cos \frac{A+B}{2}-\cos \frac{A}{2} \cos \frac{B}{2}\right)-\cos A$
$=2 \sin \frac{A}{2} \text{cosec} \frac{B}{2}\left(-\sin \frac{A}{2} \sin \frac{B}{2}\right)-\cos A$
$=-2 \sin ^{2} \frac{A}{2}-\cos A=-1$