Prove that : sin13π/3 sin 2π/3 + cos 4π/3 sin 13π/6 = 1/2

Question

Prove that:

 sin \(\cfrac{13\pi}3\) sin \(\cfrac{2\pi}3\) + cos \(\cfrac{4\pi}3\) sin \(\cfrac{13\pi}6\)  = \(\cfrac12\)

Answer 1

LHS = sin \(\cfrac{13\pi}3\) sin \(\cfrac{2\pi}3\) + cos \(\cfrac{4\pi}3\) sin \(\cfrac{13\pi}6\)

= sin 780° sin 120° + cos 240° sin 390°

= sin (90° × 8 + 60°) sin (90° × 1 + 30°) + cos (90° × 2 + 60°) sin (90° × 4 + 30°)

We know that when n is odd, sin → cos.

= sin 60° cos 30° + [-cos 60°] sin 30°

= sin 60° cos 30° - sin 30° cos 60°

We know that sin A cos B – cos A sin B = sin (A – B)

= sin (60° - 30°)

= sin 30°

= 1/2

= RHS

Hence proved.