三角函数之诱导(简化)公式 2

First

$$\begin{align} \frac{2\tan x}{\sec^{2}x}=? \\ \\ \tan x=\frac{\sin x}{\cos x}, \enspace \sec x=\frac{1}{\cos x} \\ \\ \frac{2\sin x}{\cos x}\div\frac{1}{\cos^{2}x}=\frac{2\sin x}{\cos x}\cdot\cos^{2}x \\ \\ =2\sin x\cos x \\ \\ \text{根据积化和差公式} \\ \\ 2\sin x\cos x=2\cdot\frac{\sin(x+x)+\sin(x-x)}{2} \\ \\ \Rightarrow 2\cdot\frac{\sin2x}{2} \\ \\ \Rightarrow \sin2x \end{align}$$

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Second

$$\begin{align} \frac{1-\tan^{2}x}{1+\tan^{2}x}=? \\ \\ 已知公式:\frac{1}{\cos x}=\sec x, \enspace 1+\tan^{2}x=\sec^{2} x \\ \tan x=\frac{\sin x}{\cos x} \\ \\ \Rightarrow\frac{1-\frac{\sin^{2}x}{\cos^{2}x}}{\sec^{2}x} =\frac{1}{\sec^{2}x}-\frac{ \frac{\sin^{2}x}{\cos^{2}x} }{\sec^{2}x} \\ \\ \frac{1}{\frac{1}{\cos^{2}x}} - \left(\frac{\sin^{2}x}{\cos^{2}x} \div \frac{1}{\cos^{2}x}\right) \\ \\ =\cos^{2}x-\left(\frac{\sin^{2}x}{\cos^{2}x}\cdot\cos^{2}x\right) \\ \\ =\cos^{2}x-\sin^{2}x \\ \\ \text{根据二倍角公式:} \\ \\ \Rightarrow \cos2x \end{align}$$

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