三角函数之诱导(简化)公式 2
First
\[\begin{align}
\frac{2\tan x}{\sec^{2}x}=?
\\ \\
\tan x=\frac{\sin x}{\cos x}, \enspace \sec x=\frac{1}{\cos x}
\\ \\
\frac{2\sin x}{\cos x}\div\frac{1}{\cos^{2}x}=\frac{2\sin x}{\cos x}\cdot\cos^{2}x
\\ \\
=2\sin x\cos x
\\ \\
\text{根据积化和差公式}
\\ \\
2\sin x\cos x=2\cdot\frac{\sin(x+x)+\sin(x-x)}{2}
\\ \\
\Rightarrow
2\cdot\frac{\sin2x}{2}
\\ \\
\Rightarrow \sin2x
\end{align}
\]
Second
\[\begin{align}
\frac{1-\tan^{2}x}{1+\tan^{2}x}=?
\\ \\
已知公式:\frac{1}{\cos x}=\sec x, \enspace 1+\tan^{2}x=\sec^{2} x
\\
\tan x=\frac{\sin x}{\cos x}
\\ \\
\Rightarrow\frac{1-\frac{\sin^{2}x}{\cos^{2}x}}{\sec^{2}x}
=\frac{1}{\sec^{2}x}-\frac{ \frac{\sin^{2}x}{\cos^{2}x} }{\sec^{2}x}
\\ \\
\frac{1}{\frac{1}{\cos^{2}x}} -
\left(\frac{\sin^{2}x}{\cos^{2}x}
\div
\frac{1}{\cos^{2}x}\right)
\\ \\
=\cos^{2}x-\left(\frac{\sin^{2}x}{\cos^{2}x}\cdot\cos^{2}x\right)
\\ \\
=\cos^{2}x-\sin^{2}x
\\ \\
\text{根据二倍角公式:}
\\ \\
\Rightarrow \cos2x
\end{align}
\]
