Question

If \cos\theta = \frac{2}{\sqrt{14}}cosθ= 14 ​ 2 ​ and angle \thetaθ is in Quadrant I, what is the exact value of \tan 2\thetatan2θ in simplest radical form?

Answer 1

θ is in quadrant I, so cos(θ) and sin(θ) and hence tan(θ) are all positive.

Recall that

sec²(θ) = 1 + tan²(θ)

We have

cos(θ) = 2/√14 ⇒ sec(θ) = √14/2 = √(7/2)

Then

tan(θ) = +√(sec²(θ) - 1) = √(5/2)

Also recall the identities

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

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

from which we get

tan(x + y) = (sin(x) cos(y) + cos(x) sin(y)) / (cos(x) cos(y) - sin(x) sin(y))

tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) tan(y))

so that

tan(2θ) = 2 tan(θ) / (1 - tan²(θ)) = -(2√10)/3