Hello, me again

I'm stuck on another identity:

$\displaystyle \dfrac{1 - \sin\theta}{\cos\theta} \equiv \dfrac{1}{\sec\theta + \tan\theta}$

I got:

$\displaystyle LHS \equiv \dfrac{1}{\cos\theta} - \dfrac{\sin\theta}{\cos\theta} \equiv \sec\theta - \tan\theta$

I tried starting with RHS and couldn’t do it that way either. I'd really appreciate some help.

Thank you