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Q.
If $sin^{-1} a + sin^{-1}b + sin^{-1}c = \pi $, then find the value of $a\sqrt{1-a^{2}}+b\sqrt{1-b^{2}}+c\sqrt{1-c^{2}}.$
Inverse Trigonometric Functions
Solution:
Let $sin^{-1} a = x \therefore a = sin\, x$
$sin^{-1} b = y \therefore b = sin\, y$
$sin^{-1} c = z \therefore c = sin\, z$
$\therefore a\sqrt{1-a^{2}}+b\sqrt{1-b^{2}}+c\sqrt{1-c^{2}}$
$= sin\, x cos\, x + sin \,
y cosy + sinz cosz$$=(1/2) (sin2x + sin2y + sin\, 2z) = (1/2) (4sin\,x \,sin\, y\, sin\, z)$
$= 2 \,sinx \,siny \,sinz$
$= 2abc$