Question

Why are the two graph `f(x)=tan(arccos(x/2))` and `g(x)=sqrt(4-x^2)/x` equal?

Answer 1

Please see below.

Explanation:

We have `f(x)=tan(arccos(x/2))` and let `arccos(x/2)=u`

then `cosu=x/2` which leads to

`tanu=sqrt(tan^2u)=sqrt(sec^2u-1)=sqrt(1/cos^2u-1)`

= `sqrt(1/(x^2/4)-1)=sqrt(4/x^2-1)=sqrt((4-x^2)/x^2)=sqrt(4-x^2)/x`

Hence `f(x)=tan(arccos(x/2))=sqrt(4-x^2)/x`

Hence `f(x)=tan(arccos(x/2))` and `g(x)=sqrt(4-x^2)/x` are same graph.