Question

Prove: tan2 θ•cos2 θ + cos2 θ = 1.

Answer 1

I assume you mean those 2's to be exponents. Alright, so we have:
\[\tan^{2} \theta \cos^{2} \theta + \cos^{2} \theta = 1\] Well, seeing a bunch of 2nd powers and that 1 there, my first thought is to think of a very common identity you may have seen:
\[\sin^{2} \theta + \cos^{2} \theta = 1\]
Looking at the equation, I already have the 1 and the cos^2(theta). All Im missing is a sin^2(theta) in order to fit the identity. Well, you may recall that tanx is simply the division of sinx/cosx. Knowing that, we can turn tangent into sines and cosines and see what we get:
\[\frac{\sin^{2} \theta}{\cos^{2} \theta}*\cos^{2} \theta + \cos^{2} \theta = 1 \implies \sin^{2} \theta + \cos^{2} \theta = 1\]
Transforming tan^2 into sin^2/cos^2 allowed me to cancel out cos^2 terms that appeared on the top and bottom of the fraction given. That allowed me to reduce it down to sin^2 + cos^2 = 1, which is our common pythagorean identity. So from there, we're simply allowed to say that sin^2(theta) + cos^2(theta) = 1 and therefore 1 = 1, which is a true statement and the identity is proven.