1. ## tan identity

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

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

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

thanks

2. Hello, biker.josh07!

I have a "visual" solution.

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

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

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

Consider this right triangle:

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

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

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

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

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