The value of cot⁻¹ ( (√(1 + tan²(2)) - 1)/tan(2) ) - cot⁻¹ ( (√(1 + tan²(1/2)) + 1)/tan(1/2) ) is equal to

Solution

Maths sir · 4 steps

Turn each fraction into a half-angle cotangent, but check the sign of cos 2 before you take the root.

1. Half-angle form. Here √(1+tan² x) = |sec x|, not just sec x, so the sign of cos x matters. Once the root is opened, multiply top and bottom by cos x and use the half-angle identity below.

   (1+cos x)/(sin x) = (2 cos² x/2)/(2 sin x/2 cos x/2) = cot x/2

2. First term. The angle 2 rad lies between π/2 ≈ 1.57 and π ≈ 3.14, so cos 2 < 0 and |sec 2| = -sec 2. The fraction becomes (-sec 2 - 1)/(tan 2) = -(1+cos 2)/(sin 2) = -cot 1. So the term is cot⁻¹(-cot 1) = π - cot⁻¹(cot 1) = π - 1, because 1 lies in (0, π).
3. Second term. Here cos 1/2 > 0, so the root is sec 1/2. The fraction is (1+cos 1/2)/(sin 1/2) = cot 1/4. Since 1/4 lies in (0, π), the term is 1/4.
4. So. The value is (π - 1) - 1/4 = π - 5/4, which is option (c).

The idea to keep. √(1+tan² x) is |sec x|. Find the quadrant of the angle first, then use cot⁻¹(-y) = π - cot⁻¹ y to bring the answer into (0, π).

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