一条曲线的方程为 $y=\frac{\tan x}{5+2\sin x}$。

(a) 将 $\frac{\mathrm{d}y}{\mathrm{d}x}$ 表示为用 $\sin x$ 表示的最简分式。

(b) 证明该曲线没有驻点。

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。

先用商法则，再把正切和正割写成正弦、余弦，通分后使用平方关系。

By the quotient rule,

Using $\tan x=\sin x/\cos x$ and $\sec^2x=1/\cos^2x$,

用正弦的范围给分子估计下界，同时检查曲线定义域内分母为正。

Since $-1\leqslant\sin x\leqslant1$,

Thus the numerator in (a) is never zero. On the domain of the curve, $\cos x\ne0$, so $1-\sin^2x>0$; also $5+2\sin x\geqslant3>0$.

Therefore $\mathrm{d}y/\mathrm{d}x>0$ at every point of the curve, and $\boxed{\text{there are no stationary points}}$.