Let us solve the expression [tex]\(\csc^2 \theta - 4 \cot^2 \theta\)[/tex] step-by-step.
First, let's recall the trigonometric identities for cosecant ([tex]\(\csc\)[/tex]) and cotangent ([tex]\(\cot\)[/tex]):
1. [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex]
2. [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]
Now, we need to find [tex]\(\csc^2 \theta\)[/tex] and [tex]\(\cot^2 \theta\)[/tex]:
[tex]\[
\csc^2 \theta = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta}
\][/tex]
[tex]\[
\cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta}
\][/tex]
Next, we will use these squared terms in our given expression:
[tex]\[
\csc^2 \theta - 4 \cot^2 \theta = \frac{1}{\sin^2 \theta} - 4 \left(\frac{\cos^2 \theta}{\sin^2 \theta}\right)
\][/tex]
Now, let's combine the terms over a common denominator ([tex]\(\sin^2 \theta\)[/tex]):
[tex]\[
\frac{1}{\sin^2 \theta} - 4 \cdot \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1 - 4 \cos^2 \theta}{\sin^2 \theta}
\][/tex]
This gives us the final simplified expression:
[tex]\[
\csc^2 \theta - 4 \cot^2 \theta = \frac{1 - 4 \cos^2 \theta}{\sin^2 \theta}
\][/tex]
