(a) 证明恒等式 $\sin4x\equiv4\sin x(2\cos^3x-\cos x)$。

(b) 由此求 $\displaystyle\int_0^{\frac14\pi}\cos^3x\sin4x\,\mathrm{d}x$ 的精确值。

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。

连续用两次二倍角公式，再把 $\cos2x$ 写成仅含余弦的形式。

Using the double-angle identities,

as required.

承接(a)，让被积函数出现 $\sin x$ 乘余弦的幂，然后用链式法则的逆过程积分。代入端点时注意根号的幂次。

Using the identity from (a),

With $u=\cos x$ and $\mathrm{d}u=-\sin x\,\mathrm{d}x$, the limits become $u=1$ and $u=\sqrt2/2$. Thus

Equivalently, integrating in terms of $\cos x$,

At $x=\pi/4$,

At $x=0$, the bracket is $-8/7+4/5=-12/35$. Therefore