If y(x) = cot ⁻¹( (√(1 + sin x) + √(1 - sin x))/(√(1 + sin x) - √(1 - sin x)) ),x ∈ ( π/2,π ), then dy/dx at x = 5π/6 is :

Solution

Maths sir · 4 steps

Write 1 ± sin x as perfect squares in x/2, and watch the sign of the second root here.

1. Perfect squares. Since sin² x/2 + cos² x/2 = 1 and sin x = 2 sin x/2 cos x/2, we get 1 ± sin x = (cos x/2 ± sin x/2)².
2. Take the roots. For x ∈ (π/2, π), x/2 ∈ (π/4, π/2), so sin x/2 > cos x/2 > 0. Hence √(1+sin x) = cos x/2 + sin x/2 but √(1-sin x) = sin x/2 - cos x/2, because a square root is never negative.
3. Simplify the ratio. The numerator becomes 2 sin x/2 and the denominator becomes 2 cos x/2, so the fraction is tan x/2.

   y = cot⁻¹(tan x/2) = π/2 - tan⁻¹(tan x/2) = π/2 - x/2

4. So. The last equality holds because x/2 lies in (-π/2, π/2). Then dy/dx = -1/2 for every x in the interval, including x = 5π/6. The answer is (a).

The idea to keep. Whenever you see √(1 ± sin x), convert to half-angles and then use the given interval to decide which of sin x/2 or cos x/2 is larger before removing the root.

Answer from the NTA final answer key. Solution by PrepWiser.