Question

value of cos⁴A-sin⁴A​

Answer 1

$\cos {}^{4} \alpha - \sin {}^{4} \alpha \\ \\ = (cos {}^{2} \alpha ) {}^{2} - (sin {}^{2} \alpha ) {}^{2} \\ \\ = (cos {}^{2} \alpha - sin {}^{2} \alpha )(cos {}^{2} + sin {}^{2} \alpha ) \\ \\ = (cos {}^{2} \alpha - sin {}^{2} \alpha )(1) \\ \\ = (1 - sin {}^{2} \alpha - sin {}^{2} \alpha ) \\ \\ = (1 - 2sin {}^{2} \alpha )$

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Answer 2

cos^4A = (cos^2A)^2

sin^4A = (sin^2A)^2

cos^4A - sin^4A

= (cos^2A)^2 - (sin^2A)^2

= (cos^2A + sin^2A) (cos^2A - sin^2A)

= (1) × [cos^A - (1-cos^A)]

= cos^A - 1 + cos^A

= 2cos^A - 1