Question

sin⁻¹(2x/1+x²)= 2tan⁻¹x, | x | ≤ 1,prove it

Answer 1

we have to prove that,

sin^-1[2x/(1 + x²)] = 2tan^-1x , where |x| ≤ 1

Let 2tan^-1x = A .......(1)

⇒tan^-1x = A/2

⇒tan(A/2) = x ......(2)

we know from formula

• sin2θ = 2tanθ/(1 + tan²θ)

so, sinA = 2tan(A/2)/{1 + tan²(A/2)}

from equation (2),

= 2x/(1 + x²)

hence, sinA = 2x/(1 + x²)

⇒sin^-1[2x/(1 + x²) ] = A .........(3)

from equations (1) and (3),

sin^-1[2x/(1 + x²) ] = 2tan^-1x [proved]