Question:
Evaluate
\[
1+\sec^2x\sin^2x=
\]
Evaluate
\[
1+\sec^2x\sin^2x=
\]
Show Hint
Remember the important identity:
\[
1+\tan^2x=\sec^2x
\]
It is frequently used to simplify trigonometric expressions.
Updated On: Jun 22, 2026
- \(\sin2x\)
- \(\sin^2x\)
- \(\tan^2x\)
- \(\sec^2x\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Rewrite \(\sec^2x\).
We know that \[ \sec x=\frac{1}{\cos x} \] Therefore, \[ \sec^2x=\frac{1}{\cos^2x} \] Hence, \[ \sec^2x\sin^2x = \frac{\sin^2x}{\cos^2x} \] \[ =\tan^2x \] So the given expression becomes \[ 1+\tan^2x \]
Step 2: Use the standard trigonometric identity.
We know that \[ 1+\tan^2x=\sec^2x \] Therefore, \[ 1+\sec^2x\sin^2x = \sec^2x \]
Step 3: Final conclusion.
Hence, \[ \boxed{\sec^2x} \] which corresponds to option (4).
We know that \[ \sec x=\frac{1}{\cos x} \] Therefore, \[ \sec^2x=\frac{1}{\cos^2x} \] Hence, \[ \sec^2x\sin^2x = \frac{\sin^2x}{\cos^2x} \] \[ =\tan^2x \] So the given expression becomes \[ 1+\tan^2x \]
Step 2: Use the standard trigonometric identity.
We know that \[ 1+\tan^2x=\sec^2x \] Therefore, \[ 1+\sec^2x\sin^2x = \sec^2x \]
Step 3: Final conclusion.
Hence, \[ \boxed{\sec^2x} \] which corresponds to option (4).
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