Simplify the expression by removing the trigonometric functi

Simplify the expression by removing the trigonometric functions: tan(2cos-1x)

Solution

simplify :tan[2cos^-1x]

let y = 2cos^-1x

cos(y/2) = x

sin(y/2) = sqrt(1-x^2)

So, siny = 2sin(y/2)cos(y/2) = 2xsqrt(1-x^2)

cosy = 2cos^2(y/2) -1 = 2x^2 -1

tany = 2x*sqrt(1-x^2)/(2x^2 -1)

y = tan^-1[2x*sqrt(1-x^2)/(2x^2 -1)]

So, tan[2cos^-1x] = tan[ tan^-1[2x*sqrt(1-x^2)/(2x^2 -1)] ]

= 2x*sqrt(1-x^2)/(2x^2 -1)