Sin^3x-cos^3x=(1+sinxcosx)(sinx-cosx)

1.	Sin^3x-cos^3x=(1+sinxcosx)(sinx-cosx)
Answer» To prove: $\sin^3 x-\cos^3 x=(1+\sin x \cos x)(\sin x- \cos x)$ Verification It is an identity. It can be shown by two identities. $\sin^2 x+\cos^2 x =1$ , TRIGONOMETRIC Identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ , ALGEBRAIC Identity $\text{LHS}$ $=\sin^3 x-\cos^3 x$ $=(\sin x -\cos x )(\sin^2 x +\sin x\cos x +\cos^2 x)$ [Algebraic] $=(\sin x -\cos x )(1 +\sin x\cos x)$ [Trigonometric] $=\text{RHS}$ Therefore it is verified. More Information $\sin^2 x+\cos^2 x =1$ can show different identities for other trigonometric RATIOS. (I) Dividing by $\sin^2 x$ $\implies 1+\tan^2 x =\csc^2 x$ (II) Dividing by $\cos^2 x$ $\implies 1+\cot^2 x =\sec^2 x$

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