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受欢迎的三角函数 >

证明 cos^2(x)(1-sec^2(x))=-sin^2(x)

解答

证明

cos^{2} (x) (1 - sec^{2} (x)) = - sin^{2} (x)

解答

真

求解步骤

cos^{2} (x) (1 - sec^{2} (x)) = - sin^{2} (x)

调整左侧

cos^{2} (x) (1 - sec^{2} (x))

用 sin, cos 表示

(1 - sec^{2} (x)) cos^{2} (x)

使用基本三角恒等式:

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

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

化简

(1 - (\frac{1}{cos ( x )})^{2}) cos^{2} (x) : cos^{2} (x) - 1

(1 - (\frac{1}{cos ( x )})^{2}) cos^{2} (x)

(\frac{1}{cos ( x )})^{2} = \frac{1}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( x )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( x )}

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

化简

1 - \frac{1}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) - 1}{cos ^{2} ( x )}

1 - \frac{1}{cos ^{2} ( x )}

将项转换为分式:

1 = \frac{1 cos ^{2} ( x )}{cos ^{2} ( x )}

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

乘以:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

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

分式相乘:

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

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

约分:

cos^{2} (x)

= cos^{2} (x) - 1

= cos^{2} (x) - 1

= - 1 + cos^{2} (x)

使用三角恒等式改写

- 1 + cos^{2} (x)

使用毕达哥拉斯恒等式:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= - sin^{2} (x)

= - sin^{2} (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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