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受欢迎的三角函数 >

证明 (cos(2x)-cos(4x))/(sin(2x)+sin(4x))=tan(x)

解答

证明

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) + sin ( 4 x )} = tan (x)

解答

真

求解步骤

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) + sin ( 4 x )} = tan (x)

调整左侧

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) + sin ( 4 x )}

使用三角恒等式改写

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) + sin ( 4 x )}

使用和差化积恒等式:

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

= \frac{- 2 sin ( \frac{2 x + 4 x}{2} ) sin ( \frac{2 x - 4 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

化简

\frac{- 2 sin ( \frac{2 x + 4 x}{2} ) sin ( \frac{2 x - 4 x}{2} )}{sin ( 2 x ) + sin ( 4 x )} : \frac{2 sin ( x ) sin ( 3 x )}{sin ( 2 x ) + sin ( 4 x )}

\frac{- 2 sin ( \frac{2 x + 4 x}{2} ) sin ( \frac{2 x - 4 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

- 2 sin (\frac{2 x + 4 x}{2}) sin (\frac{2 x - 4 x}{2}) = - 2 sin (\frac{6 x}{2}) sin (- \frac{2 x}{2})

- 2 sin (\frac{2 x + 4 x}{2}) sin (\frac{2 x - 4 x}{2})

同类项相加:

2 x + 4 x = 6 x

= - 2 sin (\frac{6 x}{2}) sin (\frac{2 x - 4 x}{2})

\frac{2 x - 4 x}{2} = - \frac{2 x}{2}

\frac{2 x - 4 x}{2}

同类项相加:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

= - 2 sin (\frac{6 x}{2}) sin (- \frac{2 x}{2})

= \frac{- 2 sin ( \frac{6 x}{2} ) sin ( - \frac{2 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 sin ( \frac{6 x}{2} ) sin ( - \frac{2 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

使用负角恒等式:

sin (- x) = - sin (x)

= - \frac{2 ( - sin ( \frac{2 x}{2} ) ) sin ( \frac{6 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

去除括号:

(- a) = - a

= - \frac{- 2 sin ( \frac{2 x}{2} ) sin ( \frac{6 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - (- \frac{2 sin ( \frac{2 x}{2} ) sin ( \frac{6 x}{2} )}{sin ( 2 x ) + sin ( 4 x )})

使用法则

- (- a) = a

= \frac{2 sin ( \frac{2 x}{2} ) sin ( \frac{6 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

数字相除:

\frac{2}{2} = 1

= \frac{2 sin ( x ) sin ( \frac{6 x}{2} )}{sin ( 2 x ) + sin ( 4 x )}

数字相除:

\frac{6}{2} = 3

= \frac{2 sin ( x ) sin ( 3 x )}{sin ( 2 x ) + sin ( 4 x )}

= \frac{2 sin ( x ) sin ( 3 x )}{sin ( 2 x ) + sin ( 4 x )}

使用和差化积恒等式:

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

= \frac{2 sin ( x ) sin ( 3 x )}{2 sin ( \frac{2 x + 4 x}{2} ) cos ( \frac{2 x - 4 x}{2} )}

化简

\frac{2 sin ( x ) sin ( 3 x )}{2 sin ( \frac{2 x + 4 x}{2} ) cos ( \frac{2 x - 4 x}{2} )} : \frac{sin ( x )}{cos ( x )}

\frac{2 sin ( x ) sin ( 3 x )}{2 sin ( \frac{2 x + 4 x}{2} ) cos ( \frac{2 x - 4 x}{2} )}

2 sin (\frac{2 x + 4 x}{2}) cos (\frac{2 x - 4 x}{2}) = 2 sin (\frac{6 x}{2}) cos (- \frac{2 x}{2})

2 sin (\frac{2 x + 4 x}{2}) cos (\frac{2 x - 4 x}{2})

同类项相加:

2 x + 4 x = 6 x

= 2 sin (\frac{6 x}{2}) cos (\frac{2 x - 4 x}{2})

\frac{2 x - 4 x}{2} = - \frac{2 x}{2}

\frac{2 x - 4 x}{2}

同类项相加:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

= 2 sin (\frac{6 x}{2}) cos (- \frac{2 x}{2})

= \frac{2 sin ( x ) sin ( 3 x )}{2 sin ( \frac{6 x}{2} ) cos ( - \frac{2 x}{2} )}

数字相除:

\frac{2}{2} = 1

= \frac{sin ( x ) sin ( 3 x )}{sin ( \frac{6 x}{2} ) cos ( - \frac{2 x}{2} )}

使用负角恒等式:

cos (- x) = cos (x)

= \frac{sin ( 3 x ) sin ( x )}{cos ( \frac{2 x}{2} ) sin ( \frac{6 x}{2} )}

数字相除:

\frac{2}{2} = 1

= \frac{sin ( 3 x ) sin ( x )}{cos ( x ) sin ( \frac{6 x}{2} )}

数字相除:

\frac{6}{2} = 3

= \frac{sin ( 3 x ) sin ( x )}{cos ( x ) sin ( 3 x )}

约分:

sin (3 x)

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

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

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

使用基本三角恒等式:

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

= tan (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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