Is $\tan^2\theta+1=\large\frac{1}{\sin^2\theta}$ a Fundamental Identity Wrote this down during class, and I am wondering if I incorrectly transcribed from the board. Is this identity true? And if so, how? 

To derived this quickly in your head, always note that
$$\cos^2(\theta) + \sin^2(\theta) = 1$$
from, here you could either divide the equation by $\cos^2(\theta)$ or $\sin^2(\theta)$,
Dividing by former, $\cos^2(\theta)$, we get,
$$1+tan^2(\theta) = \sec^2(\theta)$$
Dividing by the latter, $\sin^2(\theta)$, we get,
$$\cot^2(\theta) + 1 = \csc^2(\theta)$$
