Answer
Identity proved.
Work Step by Step
In order to prove the given identity, we simplify the left hand side $\text{LHS}$.
$\text{LHS } =\dfrac{(1-\sin v)(1-\sin v) +\cos v (\cos v)}{\cos v (1-\sin v)} \\=\dfrac{1-2 \sin v+\sin^2 v+\cos^2 v}{\cos v (1-\sin v)} $
Use the identity $\sin^2 v+\cos^2 v=1$
$=\dfrac{2-2 \sin v}{\cos v (1-\sin v)} \\= \dfrac{2 (1-\sin v)}{\cos v (1-\sin v) } \\=\dfrac{2}{\cos v}\\ =2 \sec v~~ [ \because \sec v =\dfrac{1}{\cos v} ] \\ = \text{ RHS}$
Thus the left-hand side equals the right-hand side and we have proven the identity.
