Proving that cos ⁡ ( arctan ⁡ ( 11 2 ) 3 ) = 2 5

Question

Proving that   cos  ⁡  (            arctan      ⁡      (              11        2            )        3    )  =      2          5

asafand2c · Accepted Answer

The fact you want to prove is equivalent to

\frac{\arctan (\frac{11}{2})}{3} = \arccos (\frac{2}{\sqrt{5}}),

that is,

\arctan (\frac{11}{2}) = 3 \arccos (\frac{2}{\sqrt{5}}),

that is,

\frac{11}{2} = \tan (3 \arccos (\frac{2}{\sqrt{5}})) .

The angle

\arccos (\frac{2}{\sqrt{5}})

is the angle opposite the shorter leg in a right triangle with legs 1 and 2, so

\arccos (\frac{2}{\sqrt{5}}) = \arctan (\frac{1}{2})

, and the fact you want to prove is therefore equivalent to

\tan (3 \arctan (\frac{1}{2})) = \frac{11}{2} .

Using the triple-angle formula

\tan (3 x) = \frac{3 \tan x - \tan^{3} x}{1 - 3 \tan^{2} x}

with

x = \arctan (\frac{1}{2})

, so

\tan x = \frac{1}{2}

and

\begin{aligned} \tan (3 \arctan (\frac{1}{2})) & = \frac{3 \tan x - \tan^{3} x}{1 - 3 \tan^{2} x} \\ = \frac{3 (\frac{1}{2}) - {(\frac{1}{2})}^{3}}{1 - 3 {(\frac{1}{2})}^{2}} \\ = \frac{11}{2} \end{aligned}

which is what you needed to show.

Proving that cos ⁡ ( arctan ⁡ (