用数学归纳法证明：对所有正整数 $n$，

其中 $P_n(x)$ 是一个 $n-1$ 次多项式。

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。

归纳求导后把分母统一为下一次幂，得到新多项式的递推式。还必须检查最高次项没有抵消，才能证明次数恰好为k，而不只是至多为k。

Let $f(x)=\tan^{-1}x$. For $n=1$,

where $P_1(x)=1$ has degree $0$. Thus the result holds for $n=1$.

Assume it holds for some integer $k\geqslant1$, with $P_k$ of degree $k-1$. Differentiating,

Define

This is a polynomial of degree at most $k$. If the leading coefficient of $P_k$ is $a\ne0$, the coefficient of $x^k$ in $P_{k+1}$ is

This formula also holds for $k=1$, when $P_1'=0$. Hence $P_{k+1}$ has degree exactly $k$, and the required formula holds for $n=k+1$.

The statement is true for $n=1$ and truth for $n=k$ implies truth for $n=k+1$. Therefore, by induction, it holds for every positive integer $n$.