Prove sin &#x2061;<!-- ⁡ --> 2 x + sin &#x2061;<!-- ⁡ --> 4 x + si

Garrett Black

Garrett Black

Answered question

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

Beginner2022-06-30Added 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.

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