Prove sin ⁡ 2 x + sin ⁡ 4 x + sin ⁡ 6 x...

Garrett Black

Garrett Black

Answered

2022-06-29

Prove sin 2 x + sin 4 x + sin 6 x = 4 cos x cos 2 x sin 3 x

Answer & Explanation

Dustin Durham

Dustin Durham

Expert

2022-06-30Added 31 answers

To solve this problem, we can use the identities:
sin A + sin B = 2 sin A + B 2 cos A B 2 ,
cos A + cos B = 2 cos A + B 2 cos A B 2 ,
and
sin 2 ϕ = 2 sin ϕ cos ϕ .
Going back to the question,
LHS = sin 2 x + sin 4 x + sin 6 x = 2 sin 3 x cos x + sin 6 x = 2 sin 3 x cos x + 2 sin 3 x cos 3 x = 2 sin 3 x ( cos x + cos 3 x ) = 2 sin 3 x × 2 cos 2 x cos x = 4 cos x cos 2 x sin 3 x = RHS .
Hence, proved.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?