Question

1 Prove that: 1/2 (cos 2© - cos 8©) = sin 5©. sin 3©.

©= theta​

Answer 1

$\frac{1}{2} (\cos2 \theta - \cos 8 \theta)$

$\boxed{\blue{\cos C -\cos D= -2 \: \sin( {\frac{ C+D}{2}} {) \: \sin(}\frac{C-D}{2})}}$

$\implies \frac{1}{2} [- 2\sin( \frac{2 \theta +8 \theta }{2} ) \sin( \frac{2 \theta - 8 \theta }{2} ) ]$

$\implies \frac{1}{2} [ - 2\sin( \frac{10\theta }{2} ) \sin( \frac{ - 6 \theta }{2} ) ]$

$\implies \frac{1}{2} [ - 2 \: \sin 5\theta \: ( - \sin 3 \theta ) ]$

$\implies \frac{1}{2} [ 2\: \sin 5\theta \: \sin 3 \theta ]$

$\orange{\therefore \: \sin 5\theta \: \sin 3 \theta }$

HOPE IT HELPS!!