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Part 1 of 3:

In this multi-part problem, we will use algebra to verify the identity

$\displaystyle \frac{ \sin(t) }{ 1-\cos(t) } = \frac{ 1+\cos(t) }{ \sin(t) }.$

First, using algebra we may rewrite the equation above as

$\displaystyle \sin(t) = \left( \frac{1+\cos(t)}{\sin(t)} \right) \cdot \Big($ $\Big)$