Let f(x)=(sin x+cos x-√2)/(sin x-cos x), x ∈[0, π]-{π/4}. Then f((7 π)/12) f^(′ ′)((7 π)/12) is equal to

Solution

Maths sir · 5 steps

Write top and bottom as single trig terms first; then f collapses to one tangent and the calculus is quick.

1. Combine sin and cos. Use sin x + cos x = √2 cos(x - π/4) and sin x - cos x = √2 sin(x - π/4). With t = x - π/4, the √2 cancels and f = (cos t - 1)/(sin t).
2. Half-angle form. Write cos t - 1 = -2 sin² t/2 and sin t = 2 sin t/2 cos t/2. One sin t/2 cancels, which leaves a single tangent.

   f(x) = -tan u, u = x/2 - π/8

3. Differentiate twice. Since du/dx = 1/2, we get f' = -1/2 sec² u and f'' = -1/2 sec² u tan u. So f · f'' = 1/2 sec² u tan² u.
4. Evaluate. At x = 7π/12, u = 7π/24 - 3π/24 = π/6. Then tan² u = 1/3 and sec² u = 4/3.
5. So. f f'' = 1/2 × 4/3 × 1/3 = 2/9, which is option (b).

The idea to keep. When a function mixes sin x ± cos x, turn each into a single shifted sine or cosine first. A one-term function is far easier to differentiate twice.

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