# Math Help - tan identity

1. ## tan identity

if anyone could show me how to do this that would be awesome

show that if: $
tan(2y)=\frac{1}{tan(x)}\$

then this is equivalent to the expression: $y=\frac{\Pi}{4}-\frac{x}{2}$

thanks

2. Hello, biker.josh07!

I have a "visual" solution.

Show that if: $\tan(2y)\:=\:\frac{1}{\tan x}\$

then this is equivalent to the expression: $y\:=\:\frac{\pi}{4}-\frac{x}{2}$

Since $\tan2y \:=\:\frac{1}{\tan x}$, then two angles are complementary.

Consider this right triangle:

Code:
                      *
* |
* x |
*     |
*       | a
*         |
*           |
* 2y          |
* - - - - - - - *
b

$\text{Since }\tan 2y \,=\,\frac{a}{b}\:\text{ and }\:\tan x \,=\,\frac{b}{a},\:\text{ then: }\:\tan2y \,=\,\frac{1}{\tan x}$

. . $x$ and $2y$ are in the same right triangle.

Hence: . $x + 2y \:=\:\frac{\pi}{2} \quad\Rightarrow\quad 2y \:=\:\frac{\pi}{2} - x$

Therefore: . $y \;=\;\frac{\pi}{4} - \frac{x}{2}$