Final answer:
To find the exact value of tan2x, rewrite the equation using trigonometric identities, then square both sides of the equation and simplify. Finally, use the identity tan(2θ) = 2tan(θ) / (1 - tan²(θ)) to find the value of tan2x.
Step-by-step explanation:
To find the exact value of tan2x, we first need to rewrite the equation using trigonometric identities.
Using the identity sin(θ + φ) = sin(θ)cos(φ) + cos(θ)sin(φ), we can rewrite the equation as sqrt(3)/2 * sin(x) + 1/2 * cos(x) = sqrt(3)/2 * cos(x).
Next, we can square both sides of the equation to get rid of the square root and simplify. Finally, we can use the identity tan(2θ) = 2tan(θ) / (1 - tan²(θ)) to find the value of tan2x.