Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #104

If \(\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\pi\), prove \(x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz\).

✅ Step-by-Step Solution

Final Answer: Identity proved.

Let α=arcsin x, β=arcsin y, γ=arcsin z. Each lies in [−π/2,π/2], their sum is π, and their cosines are the corresponding nonnegative square roots.
The left side is ½(sin2α+sin2β+sin2γ). Since α+β=π−γ, the first two sines sum to 2sinγ cos(α−β).
Add sin2γ=2sinγ cosγ and use cosγ=−cos(α+β). The total becomes 2sinγ[cos(α−β)−cos(α+β)]=4sinα sinβ sinγ.
Halving gives 2sinα sinβ sinγ=2xyz, proving the identity.

[0.5 marks] Let α=arcsin x, β=arcsin y, γ=arcsin z. Each lies in [−π/2,π/2], their sum is π, and their cosines are the corresponding nonnegative square roots.
[1 mark] The left side is ½(sin2α+sin2β+sin2γ). Since α+β=π−γ, the first two sines sum to 2sinγ cos(α−β).
[1 mark] Add sin2γ=2sinγ cosγ and use cosγ=−cos(α+β). The total becomes 2sinγ[cos(α−β)−cos(α+β)]=4sinα sinβ sinγ.
[0.5 marks] Halving gives 2sinα sinβ sinγ=2xyz, proving the identity.

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