If sin x + cos x = a, then sin^6 x + cos^6 x =

Question

\[ \text{If } \sin x+\cos x=a, \]

\[ \text{then } \sin^6x+\cos^6x= \]

Squaring,

\[ (\sin x+\cos x)^2=a^2 \]

\[ 1+2\sin x\cos x=a^2 \]

\[ \sin x\cos x=\frac{a^2-1}{2} \]

Now,

\[ \sin^6x+\cos^6x \]

\[ =(\sin^2x+\cos^2x)^3 -3\sin^2x\cos^2x(\sin^2x+\cos^2x) \]

\[ =1-3\sin^2x\cos^2x \]

\[ =1-3\left(\frac{a^2-1}{2}\right)^2 \]

\[ =1-\frac34(a^2-1)^2 \]

\[ \boxed{1-\frac34(a^2-1)^2} \]

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