Prove the Identity : \( \sin^6 x + \cos^6 x = 1 – 3\sin^2 x \cos^2 x \)

Solution:

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

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

\[ = (\sin^2 x+\cos^2 x)(\sin^4 x-\sin^2 x\cos^2 x+\cos^4 x) \]

\[ = \sin^4 x-\sin^2 x\cos^2 x+\cos^4 x \]

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

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

Hence proved.