First slide

Multiple and sub- multiple Angles

Question

lf A, B, C are angles of a triangle, then $2 \sin \frac{A}{2} cosec \frac{B}{2}$ $\sin \frac{C}{2} - \sin Acot \frac{B}{2} - \cos A$ is

Moderate

A

independent of A, B, C
B

function of A, B
C

function of C
D

none of these

Solution

$\begin{array}{l} 2 \sin \frac{A}{2} cosec \frac{B}{2} (\sin \frac{C}{2} - \cos \frac{A}{2} \cos \frac{B}{2}) - cosA \\ = 2 \sin \frac{A}{2} cosec \frac{B}{2} \times (\cos \frac{A + B}{2} - \cos \frac{A}{2} \cos \frac{B}{2}) - cosA \end{array}$
$\begin{matrix} = 2 \sin \frac{A}{2} cosec \frac{B}{2} (- \sin \frac{A}{2} \sin \frac{B}{2}) - \cos A \\ = - 2 \sin^{2} \frac{A}{2} - \cos A = - 1 \end{matrix}$

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