Question

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

Answer 1

`x=(\pi)/(4)` and
`x=-(3\pi)/(4)`

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`

`(sinx )/(cos x)=1`

`tan x=1`

`tan x=(sinx)/(cos x)`

`tan x=tan(\pi)/(4)`

`x=(\pi)/(4)`

Tan x is positive in I and III quadrant

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

Therefore,
`tan x=tan ((\pi)/(4)-\pi)=tan(\pi-4\pi)/(4)=tan(-3\pi)/(4)`

`x=-(3\pi)/(4)`