Question

cos(2 tan−1 x) simplify the expression

Answer 1

I think you need a triangle and use

\[\tan^{1} (x)\] to find the missing side.

tan = opp/adj so x/1 gives opp = x and adj = 1



so now you have

\[\cos(2 \theta) = \cos^2(\theta) - \sin^2(\theta)\]

looking at the triangle

\[\sin(\theta) = \frac{x}{\sqrt{x^2 +1}} ...and.... \cos(\theta) = \frac{1}{\sqrt{x^2 + 1}}\]

so your equation now becomes

\[\cos(2 \theta) = (\frac{1}{\sqrt{x^2 + 1}})^2 - (\frac{x}{\sqrt{x^2 + 1}})^2\]

you just need to simplify the equation there is a common denominator is

\[x^2 + 1\]