Certainly! Let's evaluate the given expression [tex]\( f\left(\frac{\cos^2\theta - 1}{\sin^2\theta}\right) \)[/tex].
To simplify the given expression, we start with the trigonometric identities:
[tex]\[ \cos^2\theta + \sin^2\theta = 1. \][/tex]
1. Simplify the numerator [tex]\( \cos^2\theta - 1 \)[/tex]:
Using the identity [tex]\( \cos^2\theta = 1 - \sin^2\theta \)[/tex]:
[tex]\[
\cos^2\theta - 1 = (1 - \sin^2\theta) - 1 = -\sin^2\theta.
\][/tex]
2. Simplify the whole fraction [tex]\( \frac{\cos^2\theta - 1}{\sin^2\theta} \)[/tex]:
Substitute [tex]\( \cos^2\theta - 1 \)[/tex] from the previous step:
[tex]\[
\frac{\cos^2\theta - 1}{\sin^2\theta} = \frac{-\sin^2\theta}{\sin^2\theta}.
\][/tex]
3. Simplify the division:
[tex]\[
\frac{-\sin^2\theta}{\sin^2\theta} = -1.
\][/tex]
So, we have simplified [tex]\( \frac{\cos^2\theta - 1}{\sin^2\theta} \)[/tex] to [tex]\(-1\)[/tex].
Given that the question asks us to evaluate [tex]\( f \left( \text{the simplified expression} \right) \)[/tex], and the simplified expression is [tex]\(-1\)[/tex], we get:
[tex]\[
f \left( -1 \right).
\][/tex]
Therefore, the correct evaluation of the given expression is [tex]\(-1\)[/tex].
So, the answer is [tex]\( \boxed{-1} \)[/tex].
