解方程 $\tan(x+45^\circ)=2\cot x$，其中 $0^\circ<x<180^\circ$。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

使用正切和角公式，将方程化成关于 $\tan x$ 的二次方程。解出后根据正切正负选象限，并核对原式的无定义角。

Within the stated interval, the original equation is undefined only at $x=45^\circ$. At $x=90^\circ$, it gives $-1=0$, so this is not a solution.

For the remaining possible angles, put $t=\tan x$. Using the addition formula,

Here $t\ne0,1$. Multiplying through,

Thus

The positive root gives a first-quadrant solution,

The negative root gives a second-quadrant solution,

Both are in the domain of the original equation. Hence