Question

Prove that $\frac{\cos \theta }{1+\sin \theta }=\frac{1-\sin \theta }{\cos \theta }$

Answer 1

$L.H.S.=\frac{\cos \theta }{1+\sin \theta }$

Multiplying and dividing numerator and denominator by $1-\sin \theta$.

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

$=\frac{\cos \theta \left(1-\sin \theta \right)}{1-{\sin }^2\theta }$

$=\frac{\cos \theta \left(1-\sin \theta \right)}{{\cos }^2\theta }$

$=\frac{1-\sin \theta }{\cos \theta }$

$=R.H.S.$

Hence, it is proved that $\frac{\cos \theta }{1+\sin \theta }=\frac{1-\sin \theta }{\cos \theta }$.