Prove this identity

A note to future me:

Today you were trying to do probability questions with your pathetically sad probability skills and you found that
\[\sum_{i=0}^{n-1}\frac{m}{t-i}\frac{\binom{t-m}{i}}{\binom{t}{i}}=1-\frac{\binom{t-m}{n}}{\binom{t}{n}}.\]
Dare you to prove it.

From,
Your arrogant younger self.

Here's a hint:
\[\frac{\binom{t-m}{i}}{\binom{t}{i}}=\frac{\binom{t-i}{m}}{\binom{t}{m}}.\]
Why? Can you think of a combinatorial interpretation for this? Proving it through algebra is trivial. Thinking of an interpretation may lead to some actual learning.

Bonus:
From the identity above, you also managed to find that it implies: \[\sum _{i=0}^n \binom{t-i}{m}=\binom{t+1}{m+1}-\binom{t-n}{m+1}.\] Prove this identity.

A curious identity

I was trying to do a question the other day when I found an interesting identity that is a corollary of the double angle formulae. I've searched online for its name, but it might be so trivial that it doesn't have one. Instead of revealing the identity, I'll let you figure it out for yourself:
\[\frac{1}{2^0}\cos(x_1) = \cos(x_1)\]
\[\frac{1}{2^1}\left(\cos(x_1+x_2)+\cos(x_1-x_2)\right)=\cos(x_1)\cos(x_2)\]
\[\frac{1}{2^{2}}\left(\begin{array}{c}
\phantom{+}\cos(x_{1}+x_{2}+x_{3})\\
+\cos(x_{1}+x_{2}-x_{3})\\
+\cos(x_{1}-x_{2}+x_{3})\\
+\cos(x_{1}-x_{2}-x_{3})\end{array}\right)=\cos(x_{1})\cos(x_{2})\cos(x_{3})\]
Do you see the pattern?
Can you prove the pattern (Hint: Try induction)?
Are there other similar identities involving sines and combinations of sines and cosines?

For those who are curious, the question that led me to this identity was:
Determine the value of \[\cos\left(\frac{\pi}{7}\right)\cos\left(\frac{2\pi}{7}\right)\cos\left(\frac{4\pi}{7}\right).\]

There is a simple way to do this question, and a hard way. Sadly the identity I found led to the hard way.

Need Practice for the GRE?

Have I got the site for you (see title).

Upon the grave realization that I'll have to take the Physics and Math GRE in a couple years (oh yes, I prepare YEARS in advance), I've started a site that updates daily with Math and Physics GRE questions. Come visit (link below), it'll be fun.

Let us go then, you and I.