Question

Solve the following equations.
$cosxcos3x=cos5xcos7x.$

Answer 1

$\cos x\cos 3x=\cos 5x\cos 7x$

We know that $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

Using the above identity, we can write the equation as:

$\cos 4x+\cos 2x=\cos 12x+\cos 2x=>\cos 12x-\cos 4x=0=>-2\sin 8x\times \sin 4x=0=>\sin 4x\times \sin 8x=0\therefore \sin 4x=0and\sin 8x=0=>4x=n\pi =>8x=m\pi =>x=\frac{n\pi }{4}=>x=\frac{m\pi }{8}$

$m,n$ belongs to integers.