Question

establish the identity
\\(\frac{1 - \cos\theta}{\sin\theta}+\frac{\sin\theta}{1 - \cos\theta}=2\csc\theta\\)

write the left side of the expression with a common denominator. do not expand the numerator
\\(\frac{(1 - \cos\theta)^2+\sin^{2}\theta}{\sin\theta(1 - \cos\theta)})

expand and simplify the numerator by rewriting without any parentheses.
\\(\frac{1 - 2\cos\theta+\cos^{2}\theta+\sin^{2}\theta}{\sin\theta(1 - \cos\theta)})

apply an appropriate pythagorean identity to simplify the numerator of the expression from the previous step.
\\(\frac{}{\sin\theta(1 - \cos\theta)}\\) (do not factor)

Explanation:

Step1: Find common denominator

We have \(\frac{1 - \cos\theta}{\sin\theta}+\frac{\sin\theta}{1 - \cos\theta}=\frac{(1 - \cos\theta)^2+\sin^{2}\theta}{\sin\theta(1 - \cos\theta)}\)

Step2: Expand numerator

Expand \((1 - \cos\theta)^2\) to get \(1-2\cos\theta+\cos^{2}\theta\), so the expression becomes \(\frac{1-2\cos\theta+\cos^{2}\theta+\sin^{2}\theta}{\sin\theta(1 - \cos\theta)}\)

Step3: Apply Pythagorean identity

Since \(\sin^{2}\theta+\cos^{2}\theta = 1\), the numerator \(1-2\cos\theta+\cos^{2}\theta+\sin^{2}\theta=1-2\cos\theta + 1=2-2\cos\theta\). So the expression is \(\frac{2-2\cos\theta}{\sin\theta(1 - \cos\theta)}\)

Step4: Factor numerator

Factor out 2 from the numerator: \(\frac{2(1 - \cos\theta)}{\sin\theta(1 - \cos\theta)}\)

Step5: Simplify

Cancel out the \((1 - \cos\theta)\) terms: \(\frac{2}{\sin\theta}\)

Step6: Use reciprocal identity

Since \(\csc\theta=\frac{1}{\sin\theta}\), \(\frac{2}{\sin\theta}=2\csc\theta\)

Answer:

The identity is established as shown above.