Let's verify the identity $(\csc(\theta)+\cot(\theta))(1-\cos(\theta))=\sin(\theta)$. What is a good first step?

A. $\csc(\theta)+\cot(\theta)-\cot(\theta)\cos(\theta)=\sin(\theta)$
B. $\csc(\theta)-\cot(\theta)+\csc(\theta)\cos(\theta)+\cot(\theta)\cos(\theta)=\sin(\theta)$
C. $\csc(\theta)+\cot(\theta)-\csc(\theta)\cos(\theta)-\cot(\theta)\cos(\theta)=\sin(\theta)$
D. $\csc(\theta)+\cos(\theta)-\csc(\theta)\cot(\theta)+\cot(\theta)\cos(\theta)=\sin(\theta)$
E. $\csc(\theta)+\cot(\theta)-\csc(\theta)\cos(\theta)+\cot(\theta)\cos(\theta)=\sin(\theta)$
F. None of the above

What do we get when we rewrite the left-hand side purely in terms of sine and cosine?

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

How can we simplify the previous equation?

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

How can we apply the Pythagorean Identity here?

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

How can we finish the proof?

A. After applying the double angle formula, both sides become equal.
B. After simplifying, we obtain the double angle formula for sine.
C. After applying the Pythagorean Identity again, both sides become equal.
D. After simplifying the left-hand side, both sides are equal.
E. None of the above