prove that f(x)= 2 tan^-1(x) + sin^-1 { 2x/1+x^2} is a constant for all x>or= 1. find that constant?

f(x) = 2tan^-1(x) + sin^-1 { 2x/1+x²}

Put x = tan θ

f(tan θ) = 2tan^-1(tan θ) + sin^-1 { 2tan θ/1+tan² θ}

f(tan θ) = 2tan^-1(tan θ) + sin^-1 { 2tan θ/sec² θ}

f(tan θ) = 2tan^-1(tan θ) + sin^-1 { 2sin θ cos θ}

f(tan θ) = 2θ + sin^-1 { sin 2θ }

f(tan θ) = 2θ + 2θ

f(tan θ) = 4θ

Now x = tan θ > = 1

θ = π/4 for x = 1,

so the constant is 4θ = π.

Regards,

Team,

TopperLearning.