Question
Simplify \( \sin ^{2} x \sec ^{2} x-\sec ^{2} x \) \( [?] \)
Ask by Haynes Moreno. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \(-1\).
Solution
1. Factor out the common term \(\sec^2 x\):
\[
\sin^2 x \sec^2 x - \sec^2 x = \sec^2 x (\sin^2 x - 1)
\]
2. Recognize that \(\sin^2 x - 1 = -(1 - \sin^2 x)\). Using the Pythagorean identity, \(1 - \sin^2 x = \cos^2 x\), so:
\[
\sin^2 x - 1 = -\cos^2 x
\]
3. Substitute back into the expression:
\[
\sec^2 x (\sin^2 x - 1) = \sec^2 x (-\cos^2 x) = -\sec^2 x \cos^2 x
\]
4. Recall that \(\sec^2 x = \frac{1}{\cos^2 x}\), hence:
\[
-\sec^2 x \cos^2 x = -\frac{1}{\cos^2 x} \cdot \cos^2 x = -1
\]
The simplified expression is \(-1\).
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Simplify this solution Mind Expander
Let’s break it down! We start with the expression \( \sin^{2} x \sec^{2} x - \sec^{2} x \). Notice that we can factor out \( \sec^{2} x \): \( \sec^{2} x (\sin^{2} x - 1) \). Now, using the Pythagorean identity \( \sin^{2} x + \cos^{2} x = 1 \), we find that \( \sin^{2} x - 1 = -\cos^{2} x \). Thus, substituting this back into our expression gives us: \( \sec^{2} x (-\cos^{2} x) = -\sec^{2} x \cos^{2} x \). Since \( \sec x = \frac{1}{\cos x} \), we can replace \( \sec^{2} x \) to get: \(-\sec^{2} x \cos^{2} x = -\frac{1}{\cos^{2} x} \cos^{2} x = -1\). So, the simplified expression is: \(-1\).
