Let’s say you’ve got some crazy trig equation that you have to solve… something like this…

tan^2(x) + tan(x) = 6

And you’ve got to figure out what x is. So you use all kinds of different trig identities, going around and around in circles trying to make the expression look nice. But that is getting you nowhere. How about try this. Let tan(x)=t. Then substitute. What you get is this:

tan^2(x) + tan(x) = 6

t^2+t=6

Subtract 6 from both sides, and then you’ve got a quadratic that you can solve by factoring or using the quadratic formula. You know how to solve quadratics, right? Right.

t^2+t=6

t^2+t-6=0

(t-2)(t+3)=0

t-2=0 or t+3=0

t=2 or t=-3

Now substitute tangent back in for t, take the inverse tangent of both sides, and you get you answer! This won’t work for every trig equation you have to solve, but its a handy dandy little trick you can keep in your back pocket for when you see a trig function squared.