Question

If $a=\frac{2\sin x}{1+\cos x+\sin x}$, then prove that $\frac{1-\cos x+\sin x}{1+\sin x}$ is also equal to $a$.

Answer 1

We have,

$a=\frac{2\sin x}{1+\cos x+\sin x}$

$\Rightarrow a=\frac{2\sin x(1-\cos x+\sin x)}{(1+\cos x+\sin x)(1-\cos x+\sin x)}$

$\Rightarrow a=\frac{2\sin x(1-\cos x+\sin x)}{(1+\sin x{)}^2-{\cos }^{2}x}=\frac{2\sin x(1-\cos x+\sin x)}{1+2\sin x+{\sin }^{2}x-{\cos }^{2}x}$

$\Rightarrow a=\frac{2\sin x(1-\cos x+\sin x)}{2\sin x+{\sin }^{2}x+{\sin }^{2}x}=\frac{1-\cos x+\sin x}{1+\sin x}$