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if sin ^-1x +sin^-1 y+ sin^-1 z= pie ,prove that

a)x( root 1-x²)+y( root 1- y²)+ z(root 1-z²)= 2xyz

b)x⁴+ y⁴+z⁴+ 4 x²y²z²= 2( x²y²+y²z²+ z²x²)

$L e t \sin^{- 1} x = A \Rightarrow \sin A = x S o, \cos A = \sqrt{1 - \sin^{2} A} = \sqrt{1 - x^{2}} A n d, \sin^{- 1} y = B \Rightarrow \sin B = y S o, \cos B = \sqrt{1 - \sin^{2} B} = \sqrt{1 - y^{2}} A n d, \sin^{- 1} z = C \Rightarrow \sin C = z G i v e n, \sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = π S o, A + B + C = π \Rightarrow A + B = π - C T a k i n g \cos o n b o t h s i d e s, w e h a v e, \cos (A + B) = \cos (π - C) \Rightarrow \cos A \cos B - \sin A \sin B = - \cos C \Rightarrow \sqrt{1 - x^{2}} \sqrt{1 - y^{2}} - x y = - \sqrt{1 - z^{2}} \Rightarrow \sqrt{1 - x^{2}} \sqrt{1 - y^{2}} = x y - \sqrt{1 - z^{2}} s q u a r i n g o n b o t h s i d e, w e h a v e, {(\sqrt{1 - x^{2}} \sqrt{1 - y^{2}})}^{2} = {(x y - \sqrt{1 - z^{2}})}^{2} \Rightarrow (1 - x^{2}) (1 - y^{2}) = x^{2} y^{2} + (1 - z^{2}) - 2 x y \sqrt{1 - z^{2}} \Rightarrow 1 - y^{2} - x^{2} + x^{2} y^{2} = x^{2} y^{2} + 1 - z^{2} - 2 x y \sqrt{1 - z^{2}} \Rightarrow x^{2} + y^{2} - z^{2} - 2 x y \sqrt{1 - z^{2}} = 0 \Rightarrow x^{2} + y^{2} - z^{2} = 2 x y \sqrt{1 - z^{2}} s q u a r i n g o n b o t h s i d e s, w e h a v e, {(x^{2} + y^{2} - z^{2})}^{2} = {(2 x y \sqrt{1 - z^{2}})}^{2} \Rightarrow x^{4} + y^{4} + z^{4} + 2 x^{2} y^{2} - 2 y^{2} z^{2} - 2 z^{2} x^{2} = 4 x^{2} y^{2} (1 - z^{2}) \Rightarrow x^{4} + y^{4} + z^{4} + 2 x^{2} y^{2} - 2 y^{2} z^{2} - 2 z^{2} x^{2} = 4 x^{2} y^{2} - 4 x^{2} y^{2} z^{2} \Rightarrow x^{4} + y^{4} + z^{4} - 2 x^{2} y^{2} - 2 y^{2} z^{2} - 2 z^{2} x^{2} + 4 x^{2} y^{2} z^{2} = 0 \Rightarrow x^{4} + y^{4} + z^{4} + 4 x^{2} y^{2} z^{2} = 2 x^{2} y^{2} + 2 y^{2} z^{2} + 2 z^{2} x^{2} \Rightarrow x^{4} + y^{4} + z^{4} + 4 x^{2} y^{2} z^{2} = 2 (x^{2} y^{2} + y^{2} z^{2} + z^{2} x^{2}) H e n c e p r o v e d . S i m i l a r l y, y o u c a n t r y t h e o t h e r p a r t .$

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