To simplify the expression [tex]\(1 - \frac{1}{\sin^2 x}\)[/tex], we follow these steps:
1. Express [tex]\( \frac{1}{\sin^2 x} \)[/tex] in terms of cotangent:
Recall the identity involving cotangent and sine:
[tex]\[
\cot x = \frac{\cos x}{\sin x}
\][/tex]
So,
[tex]\[
\cot^2 x = \left(\frac{\cos x}{\sin x}\right)^2 = \frac{\cos^2 x}{\sin^2 x}
\][/tex]
2. Simplify the fraction:
[tex]\[
\frac{1}{\sin^2 x}
\][/tex]
3. Substitute [tex]\(\frac{1}{\sin^2 x}\)[/tex] back into the original expression:
[tex]\[
1 - \frac{1}{\sin^2 x}
\][/tex]
4. Use the Pythagorean identity for sine and cosine:
Recall,
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Dividing each term by [tex]\(\sin^2 x\)[/tex] gives:
[tex]\[
1 + \cot^2 x = \frac{1}{\sin^2 x}
\][/tex]
Hence,
[tex]\[
\frac{1}{\sin^2 x} = 1 + \cot^2 x
\][/tex]
5. Substitute this result back into the expression:
[tex]\[
1 - \frac{1}{\sin^2 x} = 1 - (1 + \cot^2 x)
\][/tex]
6. Simplify the expression:
[tex]\[
1 - 1 - \cot^2 x = -\cot^2 x
\][/tex]
Thus, the simplified expression for [tex]\(1 - \frac{1}{\sin^2 x}\)[/tex] is:
[tex]\[
-\cot^2 x
\][/tex]
So the correct answer is:
[tex]\[
-\cot^2 x
\][/tex]
