Question

Prove that
2tan(π/4-A)÷1+tan²(π/4-A) = cos2A

Answer 1

Recall the following identities,

• 1 + tan²(x) = sec²(x)

• sin(2x) = 2 sin(x) cos(x)

• sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

as well as the following properties of sine and cosine,

• cos(-x) = cos(x)

• sin(-x) = -sin(x)

Then we have

2 tan(π/4 - A) / (1 + tan²(π/4 - A)) = 2 tan(π/4 - A) / sec²(π/4 - A)

… = 2 tan(π/4 - A) cos²(π/4 - A)

… = 2 (sin(π/4 - A) / cos(π/4 - A)) cos²(π/4 - A)

… = 2 sin(π/4 - A) cos(π/4 - A)

… = sin(2 (π/4 - A))

… = sin(π/2 - 2A)

… = sin(π/2) cos(-2A) + cos(π/2) sin(-2A)

… = cos(2A)

as required.