Question

Solve the equation sin x + cos x=cos 2x for 0 27. x

Answer 1

Answer:

$x=\frac{\pi}{4}$ and $x=-\frac{3\pi}{4}$

Step-by-step explanation:

We are given that sin x+cos x=cos 2x

We have to solve the given equation for $0\leq x\leq 2\pi$

$sin x+cos x=cos^2x-sin^2x$

Because $cos 2x=cos^2-sin^2$

$sinx+cos x=(sinx +cos x)(sinx-cos x)$

$1 =sin x-cos x$

$sin x=cos x$

$\frac{sinx }{cos x}=1$

$tan x=1$

$tan x=\frac{sinx}{cos x}$

$tan x=tan\frac{\pi}{4}$

$x=\frac{\pi}{4}$

Tan x is positive in I and III quadrant

In III quadrant angle$\theta$  replace by $\theta -\pi$

Therefore, $tan x=tan (\frac{\pi}{4}-\pi)=tan\frac{\pi-4\pi}{4}=tan\frac{-3\pi}{4}$

$x=-\frac{3\pi}{4}$