Question

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

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

Explanation:

Step1: Find common - denominator form

The left - hand side of the identity is \(\frac{1 - \cos\theta}{\sin\theta}+\frac{\sin\theta}{1 - \cos\theta}\). To get a common denominator of \(\sin\theta(1 - \cos\theta)\), we multiply the first fraction by \(\frac{1 - \cos\theta}{1 - \cos\theta}\) and the second fraction by \(\frac{\sin\theta}{\sin\theta}\).
\(\frac{(1 - \cos\theta)(1 - \cos\theta)+\sin^{2}\theta}{\sin\theta(1 - \cos\theta)}\)

Step2: Expand the numerator

Expand \((1 - \cos\theta)(1 - \cos\theta)\) using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = 1\) and \(b=\cos\theta\). So \((1 - \cos\theta)^2=1-2\cos\theta+\cos^{2}\theta\). Then the numerator is \(1-2\cos\theta+\cos^{2}\theta+\sin^{2}\theta\).
Since \(\sin^{2}\theta+\cos^{2}\theta = 1\), the numerator becomes \(1-2\cos\theta + 1=2-2\cos\theta\).
So the expression is \(\frac{2-2\cos\theta}{\sin\theta(1 - \cos\theta)}\)

Step3: Factor the numerator

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

Step4: Simplify the fraction

Cancel out the \((1 - \cos\theta)\) terms in the numerator and denominator. We get \(\frac{2}{\sin\theta}\)

Step5: Use the reciprocal identity

Since \(\csc\theta=\frac{1}{\sin\theta}\), \(\frac{2}{\sin\theta}=2\csc\theta\), which is the right - hand side of the identity.

The left - hand side of the expression with a common denominator (without expanding the numerator) is \((1 - \cos\theta)^2+\sin^{2}\theta\)

Answer:

\((1 - \cos\theta)^2+\sin^{2}\theta\)