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受欢迎的三角函数 >

证明 tan(2x)-tan(x)=(tan(x))/(cos(2x))

解答

证明

tan (2 x) - tan (x) = \frac{tan ( x )}{cos ( 2 x )}

解答

真

求解步骤

tan (2 x) - tan (x) = \frac{tan ( x )}{cos ( 2 x )}

调整左侧

tan (2 x) - tan (x)

用 sin, cos 表示

tan (2 x) - tan (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 2 x )}{cos ( 2 x )} - tan (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 2 x )}{cos ( 2 x )} - \frac{sin ( x )}{cos ( x )}

化简

\frac{sin ( 2 x )}{cos ( 2 x )} - \frac{sin ( x )}{cos ( x )} : \frac{sin ( 2 x ) cos ( x ) - sin ( x ) cos ( 2 x )}{cos ( 2 x ) cos ( x )}

\frac{sin ( 2 x )}{cos ( 2 x )} - \frac{sin ( x )}{cos ( x )}

cos (2 x), cos (x)

的最小公倍数:

cos (2 x) cos (x)

cos (2 x), cos (x)

最小公倍数 (LCM)

计算出由出现在

cos (2 x)

或

cos (x)

中的因子组成的表达式

= cos (2 x) cos (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

cos (2 x) cos (x)

对于

\frac{sin ( 2 x )}{cos ( 2 x )} :

将分母和分子乘以

cos (x)

\frac{sin ( 2 x )}{cos ( 2 x )} = \frac{sin ( 2 x ) cos ( x )}{cos ( 2 x ) cos ( x )}

对于

\frac{sin ( x )}{cos ( x )} :

将分母和分子乘以

cos (2 x)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) cos ( 2 x )}{cos ( x ) cos ( 2 x )}

= \frac{sin ( 2 x ) cos ( x )}{cos ( 2 x ) cos ( x )} - \frac{sin ( x ) cos ( 2 x )}{cos ( x ) cos ( 2 x )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( 2 x ) cos ( x ) - sin ( x ) cos ( 2 x )}{cos ( 2 x ) cos ( x )}

= \frac{sin ( 2 x ) cos ( x ) - sin ( x ) cos ( 2 x )}{cos ( 2 x ) cos ( x )}

= \frac{- cos ( 2 x ) sin ( x ) + cos ( x ) sin ( 2 x )}{cos ( 2 x ) cos ( x )}

改写为

= sin (2 x) cos (x) - cos (2 x) sin (x)

使用三角恒等式改写

sin (2 x) cos (x) - cos (2 x) sin (x)

使用角差恒等式:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= \frac{sin ( 2 x - x )}{cos ( 2 x ) cos ( x )}

同类项相加:

2 x - x = x

= \frac{sin ( x )}{cos ( 2 x ) cos ( x )}

= \frac{sin ( x )}{cos ( 2 x ) cos ( x )}

调整右侧

\frac{tan ( x )}{cos ( 2 x )}

用 sin, cos 表示

\frac{tan ( x )}{cos ( 2 x )}

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{\frac{s i n ( x )}{c o s ( x )}}{cos ( 2 x )}

化简

\frac{\frac{s i n ( x )}{c o s ( x )}}{cos ( 2 x )} : \frac{sin ( x )}{cos ( x ) cos ( 2 x )}

\frac{\frac{s i n ( x )}{c o s ( x )}}{cos ( 2 x )}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{sin ( x )}{cos ( x ) cos ( 2 x )}

= \frac{sin ( x )}{cos ( x ) cos ( 2 x )}

= \frac{sin ( x )}{cos ( 2 x ) cos ( x )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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