Proving tan(1/2 sin inverse 3/4) = 4-√7 / 3

Step 1: Use the identity sin(2x) = 2sin(x)cos(x) to find sin(x).

sin(2x) = 2sin(x)cos(x)

Let 2x = sin^-1(3/4)

sin(2(sin^-1(3/4))) = 2sin(sin^-1(3/4))cos(sin^-1(3/4))

sin(sin^-1(3/4)) = 3/4

cos(sin^-1(3/4)) = √(1 - (3/4)^2) = √7/4

sin(2(sin^-1(3/4))) = 2(3/4)(√7/4) = 3√7/8

Therefore, sin(x) = sin(1/2(sin^-1(3/4))) = (3√7/8)^1/2 = √(21)/8

Step 2: Use the identity tan(x/2) = sin(x)/(1+cos(x)) to find tan(1/2(sin^-1(3/4))).

cos(2x) = cos^2(x) - sin^2(x)

Let 2x = sin^-1(3/4)

cos^2(sin^-1(3/4)) = 1 - sin^2(sin^-1(3/4)) = 1 - (3/4)^2 = 7/16

cos(2(sin^-1(3/4))) = 2cos^2(sin^-1(3/4)) - 1 = 2(7/16) - 1 = -1/8

cos(sin^-1(3/4)) = √(1 - (3/4)^2) = √7/4

cos(1/2(sin^-1(3/4))) = √((1 + cos(sin^-1(3/4)))/2) = √((1 + √7/4)/2) = √(2 + √7)/4

tan(1/2(sin^-1(3/4))) = sin(1/2(sin^-1(3/4)))/(1 + cos(1/2(sin^-1(3/4))))
= (√(21)/8)/(1 + √(2 + √7)/4)
= (4 - √7)/3

Therefore, tan(1/2(sin^-1(3/4))) = (4 - √7)/3.