Prove the following identities:

1 + sin theta / 1 - sin theta = sec squared theta + 2 tan theta sec theta + tan squared theta


1 + sec theta / 1- sec theta = 1 - 2 cosec squared theta - 2 cot theta cosec theta

Prove the following identities:

\frac{1 + sin{\theta}}{1 - sin{\theta}} = sec^{2}{\theta} + 2 tan{\theta} sec{\theta} + tan^{2}{\theta}

I will get you started, then give me your thoughts.

Multiply the left side by \frac{1+sin{\theta}}{1+sin{\theta}}

\frac{1+sin{\theta}}{1-sin{\theta}}\cdot\frac{1+sin{\theta}}{1+sin{\theta }}

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

Now, continue? Remember what 1-sin^{2}{\theta} is equal to?