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

1 Answer
Jun 16, 2018

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).