Question

The values of the expression $\frac{{\sin }^{3}y}{1+\cos y}+\frac{{\cos }^{3}y}{1-\sin y}$ are :

• A. $\sqrt{2}\sin (\frac{\pi }{4}+y)$
• B. $\sqrt{2}\cos (\frac{\pi }{4}-y)$
• C. $\sqrt{2}\sin (\frac{\pi }{4}-y)$
• D. $\sqrt{2}\cos (\frac{\pi }{4}+y)$.

Answer 1

Answer: (A), (B)

$\sqrt{2}\sin (\frac{\pi }{4}+y)$
$\sqrt{2}\cos (\frac{\pi }{4}-y)$

$\frac{{\sin }^{3}y}{1+\cos y}+\frac{{\cos }^{3}y}{1-\sin y}$
$\Rightarrow \frac{{\sin }^{3}y(1-\cos y)}{1-{\cos }^{2}y}+{\cos }^{3}y(1+\sin y)1-{\sin }^{2}y$
$\Rightarrow \sin y(1-\cos y)+\cos y(1+\sin y)\Rightarrow \sin y+\cos y$
$\Rightarrow \sqrt{\sin }(\frac{\pi }{4}+y)$ or $\sqrt{2}\cos (\frac{\pi }{4}-y)$