If sin x + cos x = a, find the value of $s i n^{6} x + c o s^{6} x$ .

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Solution

sin x + cos x = a

squaring above equation gives $a^{2} - 1$

= sin 2x = 2 sin x cos x

$(s i n^{2} x + c o s^{2} x)^{3} = s i n^{6} x + c o s^{6} x + 3 s i n^{2} x c o s^{2} x 1 = s i n^{6} x + c o s^{6} x + \frac{3}{4} (2 s i n x c o s x)^{2} So, s i n^{6} x + c o s^{6} x = 1 - \frac{3}{4} (a^{2} - 1)^{2}$

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