(a) 证明 $(\cos x+\sin x)(\operatorname{cosec}x-\sec x)\equiv2\cot2x$。

(b) 由此解方程 $(\cos x+\sin x)(\operatorname{cosec}x-\sec x)=3$，其中 $0\le x\le\pi$。

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。

展开乘积后，中间的两项抵消。把剩下的余切与正切合并，再用二倍角公式。

Where the original expression is defined, $\sin x\cos x\ne0$. Expanding and simplifying,

也可先将第二个括号通分，再把分子写成平方差。

Using reciprocal identities,

The identity holds on the common domain $\sin x\cos x\ne0$.

利用 (a) 把原方程转化为余切方程。$2x$ 的范围是从 $0$ 到 $2\pi$，需保留两组解并检查原式有定义。

By part (a), the equation is $2\cot2x=3$, so $\tan2x=2/3$. In $0\le2x\le2\pi$, this gives

Both have $\sin x\cos x\ne0$. Hence

or $x=0.294$ and $1.86$ to three significant figures.