If x
Question:

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

Solution:

Disclaimer: There is some error in the given question.

The question should have been

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 $a .$

So, the solution is done accordingly.

Solution:

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

Rationalising the denominator :

$\frac{2 \sin x}{1+\sin x+\cos x} \times \frac{(1+\sin x)-\cos x}{(1+\sin x)-\cos x}$

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

 

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

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

 

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

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

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

Hence proved.

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