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? 

It is not true. For example, let $\theta=\frac{\pi}{6}$ ($30$ degrees). Then $\tan^2\theta+1=\frac{1}{3}+1=\frac{4}{3}$ while $\frac{1}{\sin^2\theta}=4$. 
But $\tan^2\theta+1=\frac{1}{\cos^2\theta}$ is true.  
To show that the identity  $\tan^2\theta+1=\frac{1}{\cos^2\theta}$ holds, recall that $\sin^2\theta+\cos^2\theta=1$ and divide both sides by $\cos^2\theta$.
Maybe what was written on the board is $\cot^2\theta+1=\frac{1}{\sin^2\theta}$. This can be proved in a way very similar to the way we proved the identity $\tan^2\theta+1=\frac{1}{\cos^2\theta}$.
Remark: Are the identities $1+\tan^2\theta=\sec^2\theta$ and $1+\cot\theta=\csc^2\theta$  fundamental? I do not think they are. For one thing, they are too close relatives of the Pythagorean Identity $\cos^2\theta+\sin^2\theta=1$, which is more basic, and more generally useful. However, $1+\tan^2\theta=\sec^2\theta$ does come up fairly often, particularly when we are integrating trigonometric functions. And $\tan\theta$ does come up naturally in geometry, and then from $1+\tan^2\theta=\sec^2\theta$ the other trigonometric functions can be calculated.
