# Solve $\mathrm{arccot}\left(\tan\frac{4\pi}{5}\right)$

Solve:

$$\mathrm{arccot}\left(\tan\frac{4\pi}{5}\right)$$

$$\mathrm{arccot}\left(\tan\frac{4\pi}{5}\right)=\mathrm{arccot}\left(\frac{1}{\cot\frac{4\pi}{5}}\right)\\\frac{1}{\cot\frac{4\pi}{5}}=\alpha$$

at the moment I don't know what to do

Let $$x=\operatorname{arccot}(\tan y)$$; then, by definition, $$\cot x=\tan y$$, so $$\cot x=\cot(\pi/2-y)$$ and therefore $$x=\frac{\pi}{2}-y+k\pi$$ with the integer $$k$$ chosen so that $$0 (the range of the cotangent function).