If the trigonometric equation, sin(πcos θ) = cos(πsin θ), then prove that cos (θ ± π/4) = 1/(2√2)
Solution: Solving the trigonometric equation
Here we have two different cases
Case 1
sin(πcos θ) = cos(πsin θ)
⇒ sin(πcos θ) = sin(π/2 – πsin θ)
⇒ πcos θ = π/2 – πsin θ
so, cos θ = 1/2 – sin θ
thus, cos θ + sin θ = 1/2
Multiply with 1/√2 then
(cos θ) × (1/√2) + (sin θ) × (1/√2) = (1/2) × (1/√2)
⇒ cos θ × cos (π/4) + sin θ × sin (π/4) = 1/(2√2)
⇒ cos (θ – π/4) = 1/(2√2)
Case 2
sin(πcos θ) = cos(πsin θ)
⇒ sin(πcos θ) = sin(π/2 + πsin θ)
⇒ πcos θ = π/2 + πsin θ
so, cos θ = 1/2 + sin θ
thus, cos θ – sin θ = 1/2
Multiply with 1/√2 then
(cos θ) × (1/√2) – (sin θ) × (1/√2) = (1/2) × (1/√2)
⇒ cos θ × cos (π/4) – sin θ × sin (π/4) = 1/(2√2)
⇒ cos (θ + π/4) = 1/(2√2)
So cos (θ ± π/4) = 1/(2√2)
