Prove that:

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Question:
Prove that: $\frac{7 \pi}{12}+\cos \frac{\pi}{12}=\sin \frac{5 \pi}{12}-\sin \frac{\pi}{12}$

Solution:

$105^{\circ}=\frac{7 \pi}{12}, 15^{\circ}=\frac{\pi}{12}, 75^{\circ}=\frac{5 \pi}{12}, 15^{\circ}=\frac{\pi}{12}$

LHS = cos105o + cos15o

= cos(90o + 15o) + cos(90o $- "> -$ 75o)

= - sin 15o + sin 75o [As cos(90o+A) = $- "> -$ sin A and cos(90o $- "> -$ B) = sin B]

= sin 75o $- "> -$ sin 15o

= RHS

Hence proved.