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If sin⁻¹ a + sin⁻¹ b + sin⁻¹ c = π, then what is the value of a√(1-a²) + b√(1-b²) + c√(1-c²)?

1. abc
2. a + b + c
3. (1)/(a)×(1)/(b)×(1)/(c)
4. 2abc

Correct answer: 2abc

Solution

Put A=asin a, B=asin b, C=asin c with A+B+C=pi. Each term a*sqrt(1-a^2)=sinA cosA=(1/2)sin2A. Sum=(1/2)(sin2A+sin2B+sin2C)=(1/2)(4 sinA sinB sinC)=2abc.

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