Question:
If \( \cos^{-1}x + \cos^{-1}y = \frac{2\pi}{7} \), then the value of \( \sin^{-1}x + \sin^{-1}y \) is equal to
If \( \cos^{-1}x + \cos^{-1}y = \frac{2\pi}{7} \), then the value of \( \sin^{-1}x + \sin^{-1}y \) is equal to
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Remember complementary inverse identities for quick simplification.
Updated On: May 8, 2026
- \( \frac{4\pi}{7} \)
- \( \frac{3\pi}{7} \)
- \( \frac{2\pi}{7} \)
- \( \frac{6\pi}{7} \)
- \( \frac{5\pi}{7} \)
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The Correct Option is
Solution and Explanation
Concept:
\[
\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}
\]
Step 1: Write expressions
\[ \sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x \] \[ \sin^{-1}y = \frac{\pi}{2} - \cos^{-1}y \]
Step 2: Add
\[ \sin^{-1}x + \sin^{-1}y = \pi - (\cos^{-1}x + \cos^{-1}y) \]
Step 3: Substitute given
\[ = \pi - \frac{2\pi}{7} \]
Step 4: Simplify
\[ = \frac{7\pi - 2\pi}{7} = \frac{5\pi}{7} \] \[ \boxed{\frac{5\pi}{7}} \]
Step 1: Write expressions
\[ \sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x \] \[ \sin^{-1}y = \frac{\pi}{2} - \cos^{-1}y \]
Step 2: Add
\[ \sin^{-1}x + \sin^{-1}y = \pi - (\cos^{-1}x + \cos^{-1}y) \]
Step 3: Substitute given
\[ = \pi - \frac{2\pi}{7} \]
Step 4: Simplify
\[ = \frac{7\pi - 2\pi}{7} = \frac{5\pi}{7} \] \[ \boxed{\frac{5\pi}{7}} \]
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