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

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

How can we use trig identities to rewrite our last equation?

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

How can we simplify the previous equation?

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

How can we finish the proof?

A. After simplifying further, we obtain the half angle formulas for sine and cosine.
B. After simplifying further, we obtain the double angle formulas for sine and cosine.
C. After simplifying further, we obtain the Pythagorean Identity.
D. After simplifying further, we obtain the double angle formulas for sine.
E. After simplifying further, we obtain the half angle formulas for cosine.
F. After simplifying further, we obtain the half angle formulas for sine.
G. After simplifying further, both sides become equal.
H. After simplifying further, we obtain the double angle formulas for cosine.
I. None of the above

(b) Evaluate tan($\frac{\pi}{8}$) exactly.

A. $-\sqrt{\frac{1-\sqrt{3}/2}{1+\sqrt{2}/2}}$
B. $\sqrt{\frac{1-\sqrt{2}/2}{1+\sqrt{2}/2}}$
C. $\sqrt{\frac{1-\sqrt{3}/2}{1+\sqrt{2}/2}}$
D. $-\sqrt{\frac{1-\sqrt{2}/2}{1+\sqrt{2}/2}}$
E. $\sqrt{\frac{1-1/2}{1+1/2}}$
F. $-\sqrt{\frac{1-\sqrt{3}/2}{1+\sqrt{3}/2}}$
G. $\sqrt{\frac{1-\sqrt{3}/2}{1+\sqrt{3}/2}}$
H. $-\sqrt{\frac{1-1/2}{1+1/2}}$
I. None of the above