The value of tan ⁻¹( (cos ( 15π/4 ) - 1)/(sin ( π/4 )) ) is equal to :

Solution

Maths sir · 4 steps

Bring 15π/4 down to π/4, and the bracket is the half-angle form of -tan π/8.

1. Reduce the angle. 15π/4 = 4π - π/4. Cosine repeats every 2π and is even, so cos 15π/4 = cos π/4.
2. Spot the half-angle. The bracket is now (cos π/4 - 1)/(sin π/4) = -(1-cos π/4)/(sin π/4). Use tan θ/2 = (1-cos θ)/(sin θ) with θ = π/4.

   (cos π/4 - 1)/(sin π/4) = -tan π/8 = -(√2-1)

3. Check the range. The principal value of tan⁻¹ lies in (-π/2, π/2). Since -π/8 is inside it, tan⁻¹(-tan π/8) = -π/8.
4. So. The value is -π/8, option (b).

The idea to keep. Whenever you see (1-cos θ)/(sin θ), write tan θ/2 at once. Then check that the angle sits in (-π/2, π/2) before cancelling tan⁻¹ with tan.

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