Question : Solve the following to find its value in terms of trigonometric ratios.
$(\sin A + \cos A)(1 - \sin A \cos A)$
Option 1: $\sin^3A+\cos^3A$
Option 2: $\sin^2A-\cos^2A$
Option 3: ${[\cos A-\sin A]\left[\sin ^2 A+\cos ^2 A\right]}$
Option 4: $\sin^3A-\cos^3A$
Correct Answer: $\sin^3A+\cos^3A$
Solution :
Given expression, $(\sin A + \cos A)(1 – \sin A \cos A)$
$\sin A+\cos A−\sin^2A\cos A-\sin A\cos^2A$
We know, $\sin^2A=1−\cos^2A$ and $\cos^2A=1−\sin^2A$
Now, $\sin A+\cos A−\sin^2A\cos A−\sin A\cos^2A$
= $\sin A+\cos A−(1−\cos^2A)\cos A−\sin A(1−\sin^2A)$
= $\sin A+\cos A−\cos A+\cos^3A−\sin A+\sin^3A$
= $\sin^3A+\cos^3A$
Hence, the correct answer is $\sin^3A+\cos^3A$.
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