Question

Prove the Following

Answer 1

\[1 + \sin \theta = \frac{ \cos^2 \theta }{ 1-\sin \theta }\]

Answer 2

Recall that \(\sin^2 \theta + \cos^2 \theta = 1\), so
\(\cos^2 \theta = 1 - \sin^2 \theta\)

So, maybe less obviously, but \(1 - \sin^2 \theta\) is a difference of squares Just think of \(\sin \theta\) as "\(x\)", so \(1-x^2 = (1-x)(1+x)\)

Thus, on the right side, you have:
\[ \frac{ \cos^2 \theta }{ 1-\sin \theta }=\frac{1-\sin^2\theta}{1-\sin\theta}=\frac{\cancel{(1-\sin\theta)}(1+\sin\theta)}{\cancel{1-\sin\theta}}\]