Question

The sum of principal solutions of $\cos 3x\cdot \cos 12x=-1$

Answer 1

$\cos 3x.\cos 12x=-1⟹\cos 15x+\cos 9x=-2$

The one and only case possible is $\cos 15x=-1$ and $\cos 9x=-1$

$⟹15x=(2n+1)\pi ⟹x=\frac{(2n+1)\pi }{15}$ and $9x=(2m+1)\pi ⟹x=\frac{(2m+1)\pi }{9}$

$⟹x=\frac{\pi }{3},\pi ,\frac{5\pi }{3}$

Sum of principal solutions is $3\pi$