### Problem Simplify $ \tan x+2 \tan 2 x+4 \tan 4 x+8 \cot 8 x $ The answer will be a trigonometric function of some simple function of $x$. #### Solution Using [[Double-or-Half Angle Identities]] formula for tangent, starting at the right. $\tan x+2 \tan 2 x+4 \tan 4 x+\frac{4\left(1-\tan ^{2} 4 x\right)}{\tan 4 x}$ $\tan x+2 \tan 2 x+\frac{4}{\tan 4 x}$ $\tan x+2 \tan 2 x+\frac{2\left(1-\tan ^{2} 2 x\right)}{\tan 2 x}$ $\tan x+\frac{2}{\tan 2 x}$ $\tan x+\frac{1-\tan ^{2} x}{\tan x}$ $\frac{1}{\tan x}$ $\boxed{\cot x}$