# Find solutions of $\cot x-\tan x=4\cot(2x)=0$ for $x \in (0°, 360°)$

Solve the equation

$$\cot x-\tan x=4\cot(2x)$$

for $0^\circ<x<360^\circ$

My attempt,

$$\frac{1}{\tan x}-\tan x=4(\frac{1}{\tan 2x})$$

$$\frac{4(1-\tan^2x)}{2\tan x}-\frac{1}{\tan x}+\tan x=0$$

$$2-2\tan^2 x-1+\tan^2x=0$$

$$1-\tan^2x=0$$

$$\tan x=\pm1$$

$$x=45^\circ,135^\circ,225^\circ,315^\circ$$

So I've checked out with Desmos which I got

It shows that the answer is $45$ and$135$.

My questions:

1)Why my $225^\circ$ and $315^\circ$ are not included in the graph? Are they incorrect?

2)If I substitute the answers back to the equation, for example,

$$4\cot (2 \cdot 45)$$ which is undefined. Why?

• Cot (90)=0 no?... – Archis Welankar Jun 1 '17 at 15:52
• I guess I took cot x=1/tanx. It's defined when I take cotx=cosx/sinx. Why? – Mathxx Jun 1 '17 at 15:55