Prove that, for any real such that , the following identity holds: x+ 3x x+ 3x =…

MATHEMATICS · Subjective · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Prove that, for any real xx such that x≠(2n+1)π2,  n∈Zx\neq \dfrac{(2n+1)\pi}{2},\;n\in\mathbb Z, the following identity holds: sin⁡x+sin⁡3xcos⁡x+cos⁡3x=tan⁡2x.\frac{\sin x+\sin 3x}{\cos x+\cos 3x}=\tan 2x.

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