Establish the following trigonometric identity for any real .

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Establish the following trigonometric identity for any real θ\theta. (i) Express sin⁡4θ+cos⁡4θ\sin^{4}\theta + \cos^{4}\theta in terms of sin⁡2θ\sin^{2}\theta and cos⁡2θ\cos^{2}\theta. (ii) Using the result of part (i) and a suitable double–angle relation, prove that sin⁡4θ+cos⁡4θ=1−12sin⁡22θ.\sin^{4}\theta + \cos^{4}\theta = 1-\frac{1}{2}\sin^{2} 2\theta.

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