First slide

Properties of ITF

Question

If $\sin^{- 1} a + \sin^{- 1} b + \sin^{- 1} c = π, then a \sqrt{1 - a^{2}} + b \sqrt{1 - b^{2}} + c \sqrt{1 - c^{2}} =$

Moderate

A

a+b+c
B

a²b²c²
C

2abc
D

4abc

Solution

Let $A = \sin^{- 1} a, B = \sin^{- 1} b, and C = \sin^{- 1} c$
we get $A + B + C = π$
$\begin{aligned} a \sqrt{1 - a^{2}} + b \sqrt{1 - b^{2}} + c \sqrt{1 - c^{2}} \\ = \frac{1}{2} (\sin 2 A + \sin 2 B + \sin 2 C) \\ = \frac{1}{2} [4 \sin Asin Bsin C] = 2 abc \end{aligned}$

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