25. Use the properties of inverse functions to evaluate the expression. cos[arcsin((2sqrt(x))/(1 + x)))

Answer 1

First you must rewrite the term of sin.

So arcsin((2sqrtx)/(1+x))

Meaning sin=((2sqrtx)/(1+x))

Sin=O/H (using SOHCAHTOA), so O=2sqrtx and H=1+x

Cos=A/H, so you must find A.

Using pythagorean theorem H^2=A^2+O^2 if you rearrange to solve for A, A=sqrt((1+x)^2-(2sqrt2)^2)

multiply out to get sqrt(1+2x+x^2-4x), simply to sqrt(x^2-2x+1), which can be factored into sqrt(x-1)^2. The sqrt and the squared cancel and you are left with A=x-1

So if Cos=A/H then Cos=(x-1)/(1+x).