solving an equation, with π, tan, and cos. help please?

Question

Write down the exact values of cos(1/6)π and tan(1/3)π (where the angles are in radians)Hence verify that x = (1/6)π is a solution of the equation:

2cosx = tan2x

please could someone help me solve this equation, thank you

Answer 1

cos (π / 6) = √3 / 2
tan (π / 6) = 1 / √3

Consider 2 cos x = tan 2x:-

2 cos x = 2 cos (π/6) = 2 x √3 / 2 = √3

tan 2x = 2 tan x / (1 - tan²x)
tan 2x = (2 / √3) / (1 - (1/3))
tan 2x = (2 / √3) x (3 / 2) = √3

Therefore 2 cos x = tan 2x = √3

Answer 2

cos pi/ 6= sqrt(3)/2
or 2 cos pi/6 = sqrt(3)

tan 2*pi/6 = tan pi/3 = sqrt(3)
clealry both are same