Question

prove that cos(π/4+x)+cos(π/4-x)=√2cosx

Answer 1

This is the answer for the given question

Answer 2

Proof:

We have been given the equation $\cos(\pi/4+x)+\cos(\pi/4-x)=\sqrt2\cos x$

We know the formula:

$]cos A+\cos B=2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$

Using this formula, the left hand side of the equation becomes

$\cos(\pi/4+x)+\cos(\pi/4-x)=2\cos(\frac{\pi/4+x+\pi/4-x}{2})\cos(\frac{\pi/4+x-\pi/4+x}{2})$

On simplifying, we get

$\cos(\pi/4+x)+\cos(\pi/4-x)=2\cos(\frac{2\pi/4}{2})\cos(\frac{2x}{2})\\\\=2cos(\pi/4)\cos x\\\\=2\cdot\frac{1}{\sqrt2}\cos x\\\\=\sqrt2\cos x\\\\\text{=Right hand side}$

Since, we got the right hand side of the expression, hence,

$\cos(\pi/4+x)+\cos(\pi/4-x)=\sqrt2\cos x$