Solve the following equations :
cos x + sin x = cos 2x + sin 2x
Ideas required to solve the problem:
The general solution of any trigonometric equation is given as –
• sin x = sin y, implies x = nπ + (– 1)ny, where n ∈ Z.
• cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.
• tan x = tan y, implies x = nπ + y, where n ∈ Z.
Given,
cos x + sin x = cos 2x + sin 2x
cos x – cos 2x = sin 2x – sin x
{∵ sin A - sin B =
∴
⇒ .
∴ .
Hence,
Either,
⇒
If tan x = tan y, implies x = nπ + y, where n ∈ Z.
∴
⇒ where m,n ϵ Z ….ans