Question

If $\frac{2\sin \theta }{1+\cos \theta +\sin \theta }=x$ find the value of $\frac{1-\cos \theta +\sin \theta }{1+\sin \theta }$

Answer 1

$\Rightarrow$ $\frac{2\sin \theta }{1+\cos \theta +\sin \theta }=x$ [ Given ] ---- ( 1 )

$\Rightarrow$ $\frac{1-\cos \theta +\sin \theta }{1+\sin \theta }$

$\Rightarrow$ $\frac{1-\cos \theta +\sin \theta }{1+\sin \theta }\times \frac{1+\cos \theta +\sin \theta }{1+\cos \theta +\sin \theta }$

$\Rightarrow$ $\frac{(1+\sin \theta {)}^2-{\cos }^2\theta }{(1+\sin \theta )(1+\cos \theta +\sin \theta )}$

$\Rightarrow$ $\frac{(1+\sin \theta )(1+\sin \theta )-\left[\right(1+\sin \theta \left)\right(1-\sin \theta \left)\right]}{(1+\sin \theta )(1+\cos \theta +\sin \theta )}$

$\Rightarrow$ $\frac{(1+\sin \theta )\left[\right(1+\sin \theta )-(1-\sin \theta \left)\right]}{(1+\sin \theta )(1+\cos \theta +\sin \theta )}$

$\Rightarrow$ $\frac{1+\sin \theta -1+\sin \theta }{1+\cos \theta +\sin \theta }$

$\Rightarrow$ $\frac{2\sin \theta }{1+\cos \theta +\sin \theta }$

$\Rightarrow$ $x$ [ From ( 1 ) ]

$\therefore$ $\frac{1-\cos \theta +\sin \theta }{1+\sin \theta }=x$