Question

Prove the identity $\sec \theta (1-\sin \theta )(\sec \theta +\tan \theta )=1$

Answer 1

L.H.S$=\sec \theta (1-\sin \theta )(\sec \theta +\tan \theta )$

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

$=\frac{1-{\sin }^2\theta }{{\cos }^2\theta }$

$=\frac{{\cos }^2\theta }{{\cos }^2\theta }=1=$R.H.S

Hence Proved.