Question

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

• A. independent of $A,B,C$
• B. function of $A,B$
• C. function of $C$
• D. none of these

Answer 1

Answer: (A)

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