2007 iTest Problems/Problem 17

Problem

If $x$ and $y$ are acute angles such that $x+y=\frac{\pi}{4}$ and $\tan{y}=\frac{1}{6}$ , find the value of $\tan{x}$ .

From the second equation, we get that $y=\arctan\frac{1}{6}$ . Plugging this into the first equation, we get:

$x+\arctan\frac{1}{6}=\frac{\pi}{4}$

Taking the tangent of both sides,

$\tan(x+\arctan\frac{1}{6})=\tan\frac{\pi}{4}=1$

From the tangent addition formula, we then get:

$\frac{\tan{x}+\frac{1}{6}}{1-\frac{1}{6}\tan{x}}=1$

$\tan{x}+\frac{1}{6}=1-\frac{1}{6}\tan{x}$ .

Rearranging and solving, we get

$\tan{x}=\boxed{\frac{5}{7}}$