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Question Number 29173 by abdo imad last updated on 04/Feb/18

$$findcos\left(5\alpha \right)intermsofcos\alpha thenfindthevalueof$$$$cos\left(\frac{\pi }{10}\right).$$

Answered by ajfour last updated on 05/Feb/18

$$\cos 5\alpha =\cos 3\alpha \cos 2\alpha -\sin 3\alpha \sin 2\alpha$$$$=(4\cos {}^3\alpha -3\cos \alpha )(2\cos {}^2\alpha -1)$$$$-(3\sin \alpha -4\sin {}^3\alpha )\left(2\sin \alpha \cos \alpha \right)$$$$=\cos \alpha (4\cos {}^2\alpha -3)(2\cos {}^2\alpha -1)$$$$-2\cos \alpha (1-\cos {}^2\alpha )(4\cos {}^2\alpha -1)$$$$=\cos \alpha (8\cos {}^4\alpha -10\cos {}^2\alpha +3$$$$+8\cos {}^4\alpha -10\cos {}^2\alpha +2)$$$$=\cos \alpha (16\cos {}^4\alpha -20\cos {}^2\alpha +5)$$$$let\alpha =\frac{\pi }{2}and\cos {}^2\alpha =t$$$$\Rightarrow 16t^2-20t+5=0$$$$16{(t-\frac{5}{8})}^2=\frac{25}{4}-5$$$$t=\cos {}^2\alpha =\frac{5}{8}+\frac{\sqrt{5}}{8}$$$$\Rightarrow \cos \alpha =\cos \frac{\pi }{10}=\sqrt{\frac{5+\sqrt{5}}{8}}.$$

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