prove that cos^6 a −sin^6 a = cos 2a (1−(1/4)sin^2 2a)


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Question Number 104894 by bramlex last updated on 24/Jul/20

$p r o v e t h a t \cos^{6} a - \sin^{6} a =$ $\cos 2 a (1 - \frac{1}{4} \sin^{2} 2 a)$
	Answered by john santu last updated on 24/Jul/20
	$(1)cos^6 a−sin^6 a = (cos^2 a−sin^2 a) (cos^4 a+sin^2 a cos^2 a+sin^4 a) =(cos^2 a−sin^2 a){(cos^2 a+sin^2 a)^2 −(1/4)(2sin a cos a)^2 } = cos 2a {1−(1/4)sin 2a } (JS ⊛)$
	$(1) \cos^{6} a - \sin^{6} a = (\cos^{2} a - \sin^{2} a)$ $(\cos^{4} a + \sin^{2} a \cos^{2} a + \sin^{4} a)$ $= (\cos^{2} a - \sin^{2} a) {{(\cos^{2} a + \sin^{2} a)}^{2} - \frac{1}{4} {(2 \sin a \cos a)}^{2}}$ $= \cos 2 a {1 - \frac{1}{4} \sin 2 a} (J S ⊛)$
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