Alright, let's solve the equation step by step to verify the given trigonometric identity:
[tex]\[
(\csc \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}
\][/tex]
First, let's express the left-hand side of the equation using basic trigonometric identities. Recall that cosecant and cotangent are defined as:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
Substitute these into the left-hand side:
[tex]\[
(\csc \theta - \cot \theta)^2 = \left(\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}\right)^2
\][/tex]
Combine the terms inside the parentheses:
[tex]\[
= \left(\frac{1 - \cos \theta}{\sin \theta}\right)^2
\][/tex]
Simplify the expression:
[tex]\[
= \frac{(1 - \cos \theta)^2}{\sin^2 \theta}
\][/tex]
Now, consider the right-hand side of the equation:
[tex]\[
\frac{1 - \cos \theta}{1 + \cos \theta}
\][/tex]
We need to determine if the simplified left-hand side equals the right-hand side. Recall the Pythagorean identity:
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Hence,
[tex]\[
\frac{(1 - \cos \theta)^2}{\sin^2 \theta} = \frac{(1 - \cos \theta)^2}{1 - \cos^2 \theta}
\][/tex]
Observe that [tex]\(1 - \cos^2 \theta\)[/tex] can be written as:
[tex]\[
1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta)
\][/tex]
So,
[tex]\[
\frac{(1 - \cos \theta)^2}{1 - \cos^2 \theta} = \frac{(1 - \cos \theta)^2}{(1 - \cos \theta)(1 + \cos \theta)}
\][/tex]
Cancel out [tex]\(1 - \cos \theta\)[/tex] in the numerator and denominator:
[tex]\[
= \frac{1 - \cos \theta}{1 + \cos \theta}
\][/tex]
Thus, we have shown that:
[tex]\[
(\csc \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}
\][/tex]
So, the identity is verified to be true.
