Question

The value of ${\sin }^3\alpha (1+\cot \alpha )+{\cos }^3\alpha (1+\tan \alpha )$ is equal to

• A. $\sin \alpha -\cos \alpha$
• B. $\sin \alpha +\cos \alpha$
• C. $\sin \alpha \cos \alpha$
• D. $co\sec \alpha \sec \alpha$

Answer 1

Answer: (B)

$\sin \alpha +\cos \alpha$

Given expression
$={\sin }^3\alpha (1+\frac{\cos \alpha }{\sin \alpha })+{\cos }^3\alpha (1+\frac{\sin \alpha }{\cos \alpha })$

$={\sin }^2\alpha (\sin \alpha +\cos \alpha )+{\cos }^2\alpha$ $(\cos \alpha +\sin \alpha )$

$=(\cos \alpha +\sin \alpha )({\sin }^2\alpha +{\cos }^2\alpha )$

$=\sin \alpha +\cos \alpha$