# How do you simplify (1 - cos^2 theta)/(1 + sin theta) = sin theta?

Nov 27, 2015

The solutions for this equation are $\theta = n \pi$, for $n \in \mathbb{Z}$.

#### Explanation:

First, it's a good idea to use the following identity:

${\cos}^{2} \theta + {\sin}^{2} \theta = 1 \textcolor{w h i t e}{\times} \iff \textcolor{w h i t e}{\times} 1 - {\cos}^{2} \theta = {\sin}^{2} \theta$

Thus, your equation can bei simplified like follows:

$\frac{{\sin}^{2} \theta}{1 + \sin \theta} = \sin \theta$

... multiply both sides with the denominator...

${\sin}^{2} \theta = \left(1 + \sin \theta\right) \sin \theta$

$\iff {\sin}^{2} \theta = \sin \theta + {\sin}^{2} \theta$

$\iff \sin \theta = 0$

If you graph the $\sin$ function, you will see that it intercepts the $x$ axis for ..., $- 2 \pi$, $- \pi$, $0$, $\pi$, $2 \pi$, $3 \pi$, ...

graph{sin x [-10, 10, -2, 2]}

So, your equation is not an identity (and thus it can't be proven as such), but it does have solutions, namely

$\theta \in n \pi$ for $n \in \mathbb{Z}$