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? 


Above, we have a right triangle with an angle $\theta$, a hypotenuse (Hyp), an opposite side (Opp), and an adjacent side (Adj). By the Pythagorean Theorem we have
$$(Adj)^2+(Opp)^2=(Hyp)^2$$
Given the following definitions
$ \begin{array}{cc}
\sin\theta=\frac{Opp}{Hyp} & \csc\theta=\frac{Hyp}{Opp} \\
\cos\theta=\frac{Adj}{Hyp} & \sec\theta=\frac{Hyp}{Adj} \\
\tan\theta=\frac{Opp}{Adj} & \cot\theta=\frac{Adj}{Opp}\end{array}$
Now consider what would happen if we divided everything in the above (Pythagorean) equation by $(Hyp)^2$.
$$\frac{(Adj)^2+(Opp)^2}{(Hyp)^2}=\frac{(Hyp)^2}{(Hyp)^2}$$
$$\frac{(Adj)^2}{(Hyp)^2}+\frac{(Opp)^2}{(Hyp)^2}=\frac{(Hyp)^2}{(Hyp)^2}$$
$$\bigg(\frac{Adj}{Hyp}\bigg)^2+\bigg(\frac{Opp}{Hyp}\bigg)^2=\bigg(\frac{Hyp}{Hyp}\bigg)$$
$$\cos^2\theta+\sin^2\theta=1$$
As an exercise, instead of dividing by $(Hyp)^2$, try dividing by $(Adj)^2$ or by $(Opp)^2$ just to see what you come up with.
After that, try taking the identity (that you just derived)
$$\cos^2\theta+\sin^2\theta=1$$
and divide both sides by of the equation by $\sin^2\theta$. Then repeat the exercise but divide both side by $\cos^2\theta$ instead.
Lastly, try experimenting with the basic definitions and manipulate them.
For example, divide the numerator and denominator of the following identity by $Hyp$.
$$\tan\theta=\frac{Opp}{Adj}=\frac{Opp/Hyp}{Adj/Hyp}=\frac{?}{?}$$
Can you also express $\tan\theta$ in terms of $\csc\theta$ and $\sec\theta$?
The best way to approach math is to attack a concept from as many different angles as possible.
The worst thing that you can do is to write down equations from the black board, and try to memorize them without understanding where they come from.
