Question

\\(\frac{\sin t}{1 - \cos t} - \csc t\\)

Explanation:

Step1: Recall csc\( t \) identity

Recall that \(\csc t=\frac{1}{\sin t}\), so rewrite the second term: \(\frac{\sin t}{1 - \cos t}-\frac{1}{\sin t}\)

Step2: Find common denominator

The common denominator of \(1 - \cos t\) and \(\sin t\) is \(\sin t(1 - \cos t)\). Rewrite each fraction:
\(\frac{\sin t\cdot\sin t}{\sin t(1 - \cos t)}-\frac{1 - \cos t}{\sin t(1 - \cos t)}\)

Step3: Combine fractions

Subtract the numerators: \(\frac{\sin^{2}t-(1 - \cos t)}{\sin t(1 - \cos t)}\)

Step4: Use Pythagorean identity

Recall \(\sin^{2}t = 1-\cos^{2}t=(1 - \cos t)(1 + \cos t)\). Substitute \(\sin^{2}t\):
\(\frac{(1 - \cos t)(1 + \cos t)-(1 - \cos t)}{\sin t(1 - \cos t)}\)

Step5: Factor numerator

Factor out \((1 - \cos t)\) from the numerator: \(\frac{(1 - \cos t)[(1 + \cos t)-1]}{\sin t(1 - \cos t)}\)

Step6: Simplify

Simplify \((1 + \cos t)-1=\cos t\), and cancel \((1 - \cos t)\) (assuming \(1-\cos t
eq0\), i.e., \(t
eq2k\pi,k\in\mathbb{Z}\)): \(\frac{\cos t}{\sin t}\)

Step7: Recall trigonometric identity

\(\frac{\cos t}{\sin t}=\cot t\)

Answer:

\(\cot t\)