Let's verify the identity $\cos(2\theta)=\frac{1-\tan^2(\theta)}{1+\tan^2(\theta)}$. What is a good first step?

A. $2\sin(\theta)\cos(\theta)=\frac{1-\tan^2(\theta)}{1+\tan^2(\theta)}$
B. $\cos^2(\theta)-\sin^2(\theta)=\frac{1-\tan^2(\theta)}{1+\tan^2(\theta)}$
C. $1-\sin^2(\theta)=\frac{1-\tan^2(\theta)}{1+\tan^2(\theta)}$
D. None of the above

In this next part, we'll use a trick: add and subtract $2\tan^2(\theta)$ to the numerator in the right-hand side. What do you get for your new right-hand side?

A. $\frac{1+\tan^2(\theta)}{1+\tan^2(\theta)}+\frac{-2\tan^2(\theta)}{1+\tan^2(\theta)}$
B. $\frac{1+\tan^2(\theta)}{1-\tan^2(\theta)}-\frac{2\tan^2(\theta)}{1-\tan^2(\theta)}$
C. $\frac{1+\tan^2(\theta)}{1-\tan^2(\theta)}$
D. $\frac{1+2\tan^2(\theta)}{1-2\tan^2(\theta)}$
E. $\frac{1-2\tan^2(\theta)}{1+\tan^2(\theta)}+\frac{+2\tan^2(\theta)}{1+\tan^2(\theta)}$
F. None of the above

What do we get when we simplify the right-hand side?

A. $\frac{\sec^2(\theta)}{1+\tan^2(\theta)}$
B. $\frac{1-2\tan^2(\theta)}{1+\tan^2(\theta)}+2$
C. $2\tan(\theta)$
D. $1-2\frac{\tan^2(\theta)}{1+\tan^2(\theta)}$
E. $\frac{\sec^2(\theta)}{1-\tan^2(\theta)}-1$
F. None of the above

What do you get when you multiply the fraction in the right-hand side by $\frac{\cos^2(\theta)}{\cos^2(\theta)}$ and simplify?

A. $\frac{\cos^2(\theta)}{\cos^2(\theta)-\sin^2(\theta)}$
B. $2\frac{\sin^2(\theta)}{\sin(\theta)\cos(\theta)}$
C. $\frac{1}{\cos^2(\theta)+\sin^2(\theta)}$
D. $\frac{\cos^2(\theta)-2\sin^2(\theta)}{\cos^2(\theta)+\sin^2(\theta)}$
E. $1-2\frac{\sin^2(\theta)}{\cos^2(\theta)+\sin^2(\theta)}$
F. None of the above

Let's simplify the previous expression and figure out what equation we are working with now. What do we have?

A. $2\sin(\theta)\cos(\theta)=\cos^2(\theta)$
B. $1-\sin^2(\theta)=2\tan(\theta)$
C. $2\sin(\theta)\cos(\theta)=1-2\sin^2(\theta)$
D. $\cos^2(\theta)-\sin^2(\theta)=2\tan(\theta)$
E. $\cos^2(\theta)-\sin^2(\theta)=1-2\sin^2(\theta)$
F. $\cos^2(\theta)-\sin^2(\theta)=\sec^2(\theta)+\sin^2(\theta)$
G. None of the above

How can we finish the proof?

A. We may add $2\sin^2(\theta)$ to both sides to obtain the Pythagorean Identity.
B. After rearranging terms, both sides are identical.
C. After rearranging terms and simplifying, we are left with the definition of secant.
D. After rearranging terms and simplifying, we are left with the double angle formula for sine.
E. None of the above