Question : Simplify the following.
$\frac{\sin^3 \alpha+\cos^3 \alpha}{\sin \alpha+\cos \alpha}$
Option 1: $1+\sin \alpha \cos \alpha$
Option 2: $\tan \alpha$
Option 3: $1-\sin \alpha \cos \alpha$
Option 4: $\sec \alpha$
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Correct Answer: $1-\sin \alpha \cos \alpha$
Solution :
$\frac{\sin^3 \alpha+\cos^3 \alpha}{\sin \alpha+\cos \alpha}$
= $\frac{(\sin \alpha+\cos \alpha)(\sin^2 \alpha+\cos^2 \alpha–\sin \alpha\cos \alpha)}{\sin \alpha+\cos \alpha}$
= $(1–\sin \alpha\cos \alpha)$ [$\because \sin^2 \alpha+\cos^2 \alpha=1]$
Hence, the correct answer is $(1–\sin \alpha\cos \alpha)$.
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