(a) 证明 $\sec^4\theta-\tan^4\theta=1+2\tan^2\theta$。

(b) 由此或使用其他方法，解方程 $\sec^42\alpha-\tan^42\alpha=2\tan^22\alpha\sec^22\alpha$，其中 $0^\circ<\alpha<180^\circ$。

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。

先作平方差分解，再用正割与正切的基本恒等式。

Using $\sec^2\theta-\tan^2\theta=1$,

where the expressions are defined.

用(a)和 $\sec^2=1+\tan^2$ 化成四次方程，再对平方变量求值。最后根据 $2\alpha$ 的完整范围列四个角。

By (a),

Cancelling the common quadratic term gives

Thus

Let $\beta=\tan^{-1}(2^{-1/4})=40.0603\ldots^\circ$. Since $0^\circ<2\alpha<360^\circ$, the angles are

Therefore

None has $\cos2\alpha=0$, so all are defined in the original equation.