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Question Number 28164 by abdo imad last updated on 21/Jan/18

$$simplify$$$$A={cos}^4\theta +{cos}^4(\theta +\frac{\pi }{4})+{cos}^4(\theta +\frac{2\pi }{4})+{cos}^4(\theta +\frac{3\pi }{4}).$$

Answered by ajfour last updated on 21/Jan/18

$$A=\cos {}^4\theta +\cos {}^4(\theta +\frac{\pi }{2})+$$$$\cos {}^4(\theta +\frac{\pi }{4})+\cos {}^4(\theta +\frac{\pi }{4}+\frac{\pi }{2})$$$$=(\cos {}^4\theta +\sin {}^4\theta )+$$$$\cos {}^4(\theta +\frac{\pi }{4})+\sin {}^4(\theta +\frac{\pi }{4})$$$$=1-2\sin {}^2\theta \cos {}^2\theta +$$$$1-2\sin {}^2(\theta +\frac{\pi }{4})\cos {}^2(\theta +\frac{\pi }{4})$$$$=2-\frac{1}{4}[2\sin {}^{2}2\theta +2\sin {}^2(\theta +\frac{\pi }{4})]$$$$=2-\frac{1}{4}[1-\cos 2\theta +1-\cos (2\theta +\frac{\pi }{2})]$$$$A=\frac{3}{2}+\left(\frac{\cos 2\theta -\sin 2\theta }{4}\right).$$

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