Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #73

For \(-1/\sqrt2\le x\le1\), prove \(\tan^{-1}\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}=\pi/4-\tfrac12\cos^{-1}x\).

✅ Step-by-Step Solution

Final Answer: Identity proved on the stated interval.

Let θ=arcsin x, so −π/4≤θ≤π/2. Both cos(θ/2)±sin(θ/2) are nonnegative.
Hence √(1+x)=cos(θ/2)+sin(θ/2) and √(1−x)=cos(θ/2)−sin(θ/2). Their difference divided by their sum is tan(θ/2).
The angle θ/2∈[−π/8,π/4] is in the arctangent principal interval, so the left side equals θ/2.
Using arcsin x+arccos x=π/2, θ/2=π/4−½arccos x, proving the identity.

[0.5 marks] Let θ=arcsin x, so −π/4≤θ≤π/2. Both cos(θ/2)±sin(θ/2) are nonnegative.
[1 mark] Hence √(1+x)=cos(θ/2)+sin(θ/2) and √(1−x)=cos(θ/2)−sin(θ/2). Their difference divided by their sum is tan(θ/2).
[1 mark] The angle θ/2∈[−π/8,π/4] is in the arctangent principal interval, so the left side equals θ/2.
[0.5 marks] Using arcsin x+arccos x=π/2, θ/2=π/4−½arccos x, proving the identity.

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