Question

Differentiate \( \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \) with respect to \( x \).

Hint:

Expressions like \( \frac{1-x^2}{1+x^2} \) often relate to \( \cos(2\tan^{-1}x) \). Use identities to simplify inverse trig problems.

Answer 1

To Differentiate:
y = cos⁻¹ ( (1 − x²) / (1 + x²) )

Step 1: Use Trigonometric Identity
We use the identity:

cos(2θ) = (1 − tan²θ) / (1 + tan²θ)

Comparing with (1 − x²) / (1 + x²), we get:

Let x = tan θ

Then,

(1 − x²) / (1 + x²) = cos(2θ)

Therefore,

y = cos⁻¹(cos(2θ))

Since x ∈ ℝ, we take principal value:

y = 2θ

But θ = tan⁻¹(x)

Hence,

y = 2 tan⁻¹(x)

Step 2: Differentiate

dy/dx = 2 × d/dx [ tan⁻¹(x) ]

= 2 × (1 / (1 + x²))

Final Answer:
dy/dx = 2 / (1 + x²)