Question

Find all solutions in the interval [0 degrees, 360 degrees):
cot^4(2x)=9

Answer 1

\[\begin{align*}\cot^42x&=9\\
\cot^42x-9&=0\\
\left(\cot^22x-3\right)\left(\cot^22x+3\right)&=0\end{align*}\]
So now it's a matter of solving the two equations,
\[\cot^22x-3=0\\
\cot^22x+3=0\]
\[\cot^22x=\pm3~~\iff~~\cot2x=\pm\sqrt3~~\text{(because the negative 3 is extraneous)}\]
So for which angles does cotangent give a positive and negative \(\sqrt3\)?