trigonometry question

Question

Verify the identity (2tanθ)/1+tan^2θ = sin2θ, given that trigonometric point P(θ)= (a,b). USE the given trigonometric point in the solution.

Answer 1

P(θ) = (cosθ, sinθ) = (a, b). So, a = cosθ and b = sinθ. Therefore, a² + b² = 1 and tanθ = sinθ/cosθ = b/a.

So, (2tanθ) / (1+tan²θ) = 2(b/a) / (1 + b²/a²)

= 2(b/a) / [(a²+b²) / a²]

= (2b/a) (a²/ (a²+b²)

= 2ab / (a²+b²)

= 2ab = 2(cosθ)(sinθ) = sin(2θ)