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受欢迎的三角函数 >

证明 cos(x)(sec(x)-1)=1-cos(x)

解答

证明

cos (x) (sec (x) - 1) = 1 - cos (x)

解答

真

求解步骤

cos (x) (sec (x) - 1) = 1 - cos (x)

调整左侧

cos (x) (sec (x) - 1)

用 sin, cos 表示

(- 1 + sec (x)) cos (x)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

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

化简

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

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

化简

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

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

将项转换为分式:

1 = \frac{1 cos ( x )}{cos ( x )}

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

乘以:

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

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

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

分式相乘:

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

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

约分:

cos (x)

= - - cos (x) + 1

= - cos (x) + 1

= 1 - cos (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 csc^2(θ)(1-cos^2(θ))=-cos(pi/2)

p ro v e csc^{2} (θ) (1 - cos^{2} (θ)) = - cos (\frac{π}{2})

证明 4sin^2(x)+4cos^2(x)=4

p ro v e 4 sin^{2} (x) + 4 cos^{2} (x) = 4

证明 cos(2x)=(2-sec^2(x))/(sec^2(x))

p ro v e cos (2 x) = \frac{2 - sec ^{2} ( x )}{sec ^{2} ( x )}

证明 cot(y)=(sin(2y))/(1-cos(2y))

p ro v e cot (y) = \frac{sin ( 2 y )}{1 - cos ( 2 y )}

证明 1+1/(tan^2(α))= 1/(sin^2(α))

p ro v e 1 + \frac{1}{tan ^{2} ( α )} = \frac{1}{sin ^{2} ( α )}