Answer
See proof
Work Step by Step
Verify the following identity:
$\frac{(\cos(u))}{(1 - \sin(u))} = \sec(u) + \tan(u)$
Multiply both sides by $1 - \sin(u)$:
$\cos(u) \stackrel{?}{=}(1 - \sin(u)) (\sec(u) + \tan(u))$
Write secant as 1/cosine and tangent as sine/cosine:
$\cos(u) \stackrel{?}{=}(1 - \sin(u)) \left(\frac{1}{\cos(u)} + \frac{\sin(u)}{\cos(u)}\right)$
Put $\frac{1}{\cos(u)} + \frac{\sin(u)}{\cos(u)}$ over the common denominator
$\cos(u) \stackrel{?}{=}(1 - \sin(u))\frac{1 + \sin(u)}{\cos(u)} $
Multiply both sides by $\cos(u)$:
$\cos^2(u) \stackrel{?}{=}(1 - \sin(u)) (1 + \sin(u))$
Use the Pythagorean identity: $\cos^2 (u) = 1 - \sin(u)^2$:
$1 -\sin^2(u) \stackrel{?}{=} 1 - \sin^2(u)$:
The left hand side and right hand side are identical, thus the identity is verified.
