Let M and m respectively be the maximum and minimum values of the function
f(x) = tan⁻¹ (sin x + cos x) in [ 0,π/2 ], then the value of tan(M - m) is equal to :

Solution

Maths sir · 4 steps

Find the range of sin x + cos x first, because tan⁻¹ only stretches it and keeps the order.

1. Range of the inside. Write sin x + cos x = √2 sin(x + π/4). For x ∈ [0, π/2], the angle x + π/4 runs from π/4 to 3π/4. So the sine runs from 1/√2 up to 1, and the inside runs from 1 to √2.
2. Apply inverse tangent. tan⁻¹ is increasing, so the largest input gives the largest output. That gives M = tan⁻¹√2 and m = tan⁻¹ 1 = π/4.

   tan(M - m) = (tan M - tan m)/(1 + tan M tan m) = (√2 - 1)/(1 + √2)

3. Rationalise. Multiply the top and bottom by √2 - 1: (√2-1)²/(2 - 1) = 2 - 2√2 + 1 = 3 - 2√2.
4. So. tan(M - m) = 3 - 2√2, which is option (d).

The idea to keep. For tan⁻¹ of an expression, find the range of the expression and push both ends through tan⁻¹, since it is increasing. Then use the tan(A - B) formula.

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