Question

How do you simplify `cos(2 tan ^-1 x)`?

Answer 1

Use double angle formula to remove coefficient inside the cos, then rearrange standard trig definitions to make the trig function match the inverse trig function inside the bracket

Explanation:

Recall the double angle formula:
`cos2theta=1-2sin^2theta`

Then `cos(2arctanx)=1-2sin^2arctanx`. NB I've written "arctan" here rather than "`tan^(-1)`" because the combination of exponents meaning powers and function inverses is potentially confusing.

So we now have a trig function of an inverse trig function. If we can express our `sin` in terms of `tan`, this will cancel right out.

By definition, `tantheta=(sintheta)/(costheta)=(sintheta)/sqrt(1-sin^2theta)`, so
`tan^2theta(1-sin^2theta)=sin^2theta`
`tan^2theta=sin^2theta(1+tan^2theta)`
`sin^2theta=tan^2theta/(1+tan^2theta)`

By definition, `tanarctanx=x`, so `1-2sin^2arctanx` becomes `1-(2x^2)/(1+x^2)`. Putting this over a common denominator makes `(1-x^2)/(1+x^2)`.

So
`cos(2arctanx)=(1-x^2)/(1+x^2)`.