Question

Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.

Options

• (1 + x)²f'(x) + 2(f(x))² = 0

• (1 – x)²f'(x) + 2(f(x))² = 0

• (1 + x)²f'(x) – 2(f(x))² = 0

• (1 – x)²f'(x) – 2(f(x))² = 0

Answer 1

Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then `underlinebb((1 - x)^2f^'(x) + 2(f(x))^2 = 0)`.

Explanation:

Put x = sin²θ, 0 < x < 1

⇒ sinθ = `sqrt(x)`

Now, f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - sin^2θ)/(sin^2θ)))`

⇒ f(x) = `cos(2tan^-1(sinθ))`

⇒ f(x) = `cos(2tan^-1sqrt(x)))` = `costan^-1((2sqrt(x))/(1 - x))`

⇒ f(x) = `(1 - x)/(1 + x)`

⇒ f'(x) = `((1 + x)(-1) - (1 - x).1)/(1 + x)^2` = `(-2)/(1 + x)^2`

⇒ f'(x).(1 – x)² = `-2((1 - x)/(1 + x))^2`

⇒ (1 – x)²f'(x) + 2(f(x))² = 0