Question

tan(x)-3cot(x)=0
(sin(x)/cos(x))-3(cos(x)/sin(x))
(sin^2(x)/(cos(x))(sin(x))-3(cos^2(x)/(cos(x))(sin(x))

so where do I go from here?

Answer 1

\[\begin{align*}\tan x-3\cot x&=0\\
\frac{\sin x}{\cos x}-3\frac{\cos x}{\sin x}&=0\\
\cos x\left(\frac{\sin x}{\cos x}-\frac{3\cos x}{\sin x}\right)&=0\cos x\\
\sin x-\frac{3\cos^2 x}{\sin x}&=0\\
\sin x\left(\sin x-\frac{3\cos^2x}{\sin x}\right)&=0\sin x\\
\sin^2x-3\cos^2x=0
\end{align*}\]
Next, use the identity \(\sin^2x+\cos^2x=1\):
\[(1-\cos^2x)-3\cos^2x=0\\
1-4\cos^2x=0\]
which you can solve by factoring:
\[(1-2\cos x)(1+2\cos x)=0\]
or by some simpler rewriting:
\[\cos^2x=\frac{1}{4}\\
\cos x=\pm\frac{1}{2}\]