Write the expression as an algebraic expression in x for x > 0.

Question

Write the expression as an algebraic expression in x for x > 0.

tan(2 cos⁻¹ 3/x)

Answer 1

Answer 2

tan(2arccos3/x)

construct a right triangle for the reference angle

arccos(3/x) has opposite side of the reference angle = sqr(x^2-3^2) = sqr(x^2-9)

adjacent side=3, hypotenuse=x

tangent of an angle = opposite side over adjacent side = (sqr(x^2-9))/3

tan(arccos3/x) = (sqr(x^2-9)/3 = tangent of half the angle you want

use double angle trig identity for tangent of twice an angle A

tan2A = 2tanA/(1-tan^2(A))

tan(2arccos3/x) = (2sqr(x^2-9)/3)/(1-(x^2-9)/3^2)

= (2sqr(x^2-9)/3)/((18-x^2)/9)

= (2sqr(x^2-9)/(18-x^2)/3

= 6sqr(x^2-9)/(18-x^2)

where 0<x<3,

you could add that x can't = sqr18 = 3sqr2 = about 3(1.4) = 4.2 but that's redundant as x<3 includes not= 3sqr2