Answer
The equation is an identity.
Please see proof in "step-by-step"
Work Step by Step
With
$\displaystyle \sec\theta=\frac{1}{\cos\theta},\quad$and $\cos(-\theta)=\cos\theta$
$ LHS=\displaystyle \frac{1}{\cos\theta}-\cos\theta,\quad$
... common denominator is $\cos\theta$
$LHS=\displaystyle \frac{1-\cos^{2}\theta}{\cos\theta}$
in the numerator, apply the Pythagorean identity
$\cos^{2}\theta+\sin^{2}\theta=1$
$LHS=\displaystyle \frac{\cos^{2}\theta+\sin^{2}\theta-\cos^{2}\theta}{\cos\theta}=\frac{\sin^{2}\theta}{\cos\theta}=RHS$