Question

Sin power 6 x + cos Power 6 x =1- 3 sin ² cos ²​ x

Answer 1

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \boxed{\boxed { \huge \mathcal\red{ solution}}}}$

TO ProVE

$\: \: \: \: sin {}^{6}x + cos {}^{6} x = 1 - 3sin {}^{2} x \: cos {}^{2} x$

PROviNG

$L.H.S. = sin {}^{6} x + \: cos {}^{6}x \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (sin {}^{2}x ) {}^{3} + (cos {}^{2} x) {}^{3} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (sin {}^{2} x + \: cos {}^{2} x)(sin {}^{4} x + cos {}^{4} x - sin {}^{2} x \: cos {}^{2} x) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = (1)((sin {}^{2} x + \: cos {}^{2} x) {}^{2} - 2sin {}^{2} x \: cos {}^{2} x) - sin {}^{2} x \: cos {}^{2} x) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = ((1) {}^{2} - 3sin {}^{2} x \: cos {}^{2} x) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = 1 - 3sin {}^{2} x \: cos {}^{2} x = R.H.S.(proved)$

Formula used

•) sin²x+cos²x=1

•) x²+y²=(x+y)²-2xy

•) x⁴+y⁴=(x²+y²)²-2x²y²

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