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

s i m p l i f y

A = {c o s}^{4} θ + {c o s}^{4} (θ + \frac{π}{4}) + {c o s}^{4} (θ + \frac{2 π}{4}) + {c o s}^{4} (θ + \frac{3 π}{4}) .

Answered by ajfour last updated on 21/Jan/18

A = \cos^{4} θ + \cos^{4} (θ + \frac{π}{2}) +

\cos^{4} (θ + \frac{π}{4}) + \cos^{4} (θ + \frac{π}{4} + \frac{π}{2})

= (\cos^{4} θ + \sin^{4} θ) +

\cos^{4} (θ + \frac{π}{4}) + \sin^{4} (θ + \frac{π}{4})

= 1 - 2 \sin^{2} θ \cos^{2} θ +

1 - 2 \sin^{2} (θ + \frac{π}{4}) \cos^{2} (θ + \frac{π}{4})

= 2 - \frac{1}{4} [2 \sin^{2} 2 θ + 2 \sin^{2} (θ + \frac{π}{4})]

= 2 - \frac{1}{4} [1 - \cos 2 θ + 1 - \cos (2 θ + \frac{π}{2})]

A = \frac{3}{2} + (\frac{\cos 2 θ - \sin 2 θ}{4}) .

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