Question:
If \( x = 5 + 2\sec \theta \) and \( y = 5 + 2\tan \theta \), then \( (x-5)^2 - (y-5)^2 \) is equal to
If \( x = 5 + 2\sec \theta \) and \( y = 5 + 2\tan \theta \), then \( (x-5)^2 - (y-5)^2 \) is equal to
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Always reduce expressions to standard trigonometric identities before simplifying.
Updated On: May 8, 2026
- \(3\)
- \(1\)
- \(0\)
- \(4\)
- \(2\)
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The Correct Option is D
Solution and Explanation
Concept:
Use identity:
\[
\sec^2 \theta - \tan^2 \theta = 1
\]
Step 1: Rewrite expression
\[ (x-5)^2 = (2\sec \theta)^2 = 4\sec^2 \theta \] \[ (y-5)^2 = (2\tan \theta)^2 = 4\tan^2 \theta \]
Step 2: Substitute
\[ (x-5)^2 - (y-5)^2 = 4\sec^2 \theta - 4\tan^2 \theta \]
Step 3: Factor out 4
\[ = 4(\sec^2 \theta - \tan^2 \theta) \]
Step 4: Apply identity
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Step 5: Final result
\[ = 4 \cdot 1 = 4 \] \[ \boxed{4} \]
Step 1: Rewrite expression
\[ (x-5)^2 = (2\sec \theta)^2 = 4\sec^2 \theta \] \[ (y-5)^2 = (2\tan \theta)^2 = 4\tan^2 \theta \]
Step 2: Substitute
\[ (x-5)^2 - (y-5)^2 = 4\sec^2 \theta - 4\tan^2 \theta \]
Step 3: Factor out 4
\[ = 4(\sec^2 \theta - \tan^2 \theta) \]
Step 4: Apply identity
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Step 5: Final result
\[ = 4 \cdot 1 = 4 \] \[ \boxed{4} \]
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