The expression Sin t / (1 - cos t) + (1 - cos t) / sin t simplifies to 1, by using the Pythagorean identity sin^2(t) + cos^2(t) = 1 to convert the terms into a single trigonometric function and cancel out like terms.
To simplify the expression Sin t / (1 - cos t) + (1 - cos t) / sin t, we should look for trigonometric identities that may help with the terms that involve sine and cosine. Let's consider the Pythagorean identity sin2(t) + cos2(t) = 1 and the fact that we can write 1 as sin2(t) + cos2(t).
By replacing the '1' in the expression with sin2(t) + cos2(t), we can simplify the expression to a single trigonometric function. After replacing and simplifying, the expression becomes:
(sin2(t) + cos2(t)) / sin t - cos t / sin t + cos t / sin t - (sin2(t) + cos2(t)) / sin t
This simplifies further as the cos t / sin t terms cancel out and we are left with 1 - cos2(t) / sin t, which is sin t / sin t = 1. Therefore, the simplified expression is 1.