Simplify the expression tan(2cos^−1(x/5))

Answer 1

use the pythagorean theorem

tan(cos^-1(x)) = [sqr(1-x^2)]/x

replace x with x/5

tan(cos^-1(x/5)) = [sqr(25-x^2)]/x

use the double angle formula for tangents

tan2x = 2tanx/(1-tan^2x)

tan[2cos^-1(x/5)] = {2[sqr(25-x^2)]/x}/[(1-(25-x^2))/x^2]

= 2x[sqr(25-x^2)]/(2x^2-25)

-5< x < 5 since the cosine of any angle is between -1 and 1

-1 < x/5 < 1