Problem #508
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508
Prove the identity \((\sin x + \cos x)(1 - \sin x \cos x) \equiv \sin^3 x + \cos^3 x\).
Solution
Start with the left-hand side: \((\sin x + \cos x)(1 - \sin x \cos x)\).
Expand the expression: \(\sin x + \cos x - \sin^2 x \cos x - \cos^2 x \sin x\).
Rearrange terms: \(\sin x + \cos x - \sin^2 x \cos x - \cos^2 x \sin x\).
Use the identities \(\sin^2 x = 1 - \cos^2 x\) and \(\cos^2 x = 1 - \sin^2 x\).
Substitute these into the expression: \(\sin x + \cos x - (1 - \cos^2 x) \cos x - (1 - \sin^2 x) \sin x\).
Simplify: \(\sin x + \cos x - \cos x + \cos^3 x - \sin x + \sin^3 x\).
Combine like terms: \(\sin^3 x + \cos^3 x\).
Thus, the identity is proven: \((\sin x + \cos x)(1 - \sin x \cos x) = \sin^3 x + \cos^3 x\).