Question:
If \( \sin^{-1} a + \sin^{-1} b + \sin^{-1} c = \pi \), then the value of \( a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2} \) will be
If \( \sin^{-1} a + \sin^{-1} b + \sin^{-1} c = \pi \), then the value of \( a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2} \) will be
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Memorize special identity for $\sin^{-1}a+\sin^{-1}b+\sin^{-1}c=\pi$.
Updated On: Apr 23, 2026
- $2abc$
- $abc$
- $\frac{1}{2}abc$
- $\frac{1}{3}abc$
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The Correct Option is A
Solution and Explanation
Concept:
Use identity:
\[
\sin^{-1}a + \sin^{-1}b + \sin^{-1}c = \pi
\Rightarrow a^2 + b^2 + c^2 + 2abc = 1
\]
Step 1: Use identity.
Step 2: Simplify expression.
Using symmetry and identity: \[ a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2} = 2abc \] Conclusion:
Answer = $2abc$
Step 1: Use identity.
Step 2: Simplify expression.
Using symmetry and identity: \[ a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2} = 2abc \] Conclusion:
Answer = $2abc$
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