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受欢迎的三角函数 >

证明 cos(2x-x)-2sin(x)sin(2x)=cos(3x)

解答

证明

cos (2 x - x) - 2 sin (x) sin (2 x) = cos (3 x)

解答

真

求解步骤

cos (2 x - x) - 2 sin (x) sin (2 x) = cos (3 x)

调整左侧

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

使用三角恒等式改写

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

使用积化和差恒等式:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= cos (2 x - x) - 2 \cdot \frac{1}{2} (cos (x - 2 x) - cos (x + 2 x))

化简

cos (2 x - x) - 2 \cdot \frac{1}{2} (cos (x - 2 x) - cos (x + 2 x)) : cos (3 x)

cos (2 x - x) - 2 \cdot \frac{1}{2} (cos (x - 2 x) - cos (x + 2 x))

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

cos (2 x - x)

同类项相加:

2 x - x = x

= cos (x)

2 \cdot \frac{1}{2} (cos (x - 2 x) - cos (x + 2 x)) = cos (x) - cos (3 x)

2 \cdot \frac{1}{2} (cos (x - 2 x) - cos (x + 2 x))

分式相乘:

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

= \frac{1 \cdot 2 ( cos ( x - 2 x ) - cos ( x + 2 x ) )}{2}

约分:

2

= 1 \cdot (cos (x - 2 x) - cos (x + 2 x))

同类项相加:

x - 2 x = - x

= 1 \cdot (cos (- x) - cos (x + 2 x))

同类项相加:

x + 2 x = 3 x

= 1 \cdot (cos (- x) - cos (3 x))

整理后得

= cos (- x) - cos (3 x)

使用负角恒等式:

cos (- x) = cos (x)

= cos (x) - cos (3 x)

= cos (x) - (cos (x) - cos (3 x))

- (cos (x) - cos (3 x)) : - cos (x) + cos (3 x)

- (cos (x) - cos (3 x))

打开括号

= - (cos (x)) - (- cos (3 x))

使用加减运算法则

- (- a) = a, - (a) = - a

= - cos (x) + cos (3 x)

= cos (x) - cos (x) + cos (3 x)

同类项相加:

cos (x) - cos (x) = 0

= cos (3 x)

= cos (3 x)

= cos (3 x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

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p ro v e 2 (cos (3 θ) + 3 cos (θ)) = (2 cos (θ))^{3}

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p ro v e \frac{sin ( x ) tan ( x )}{cos ( x ) + 1} = tan (x)

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p ro v e tan (8 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

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p ro v e \frac{cos ( x ) sin ( x )}{sin ( x ) cos ( x )} = 1