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受欢迎的三角函数 >

证明 sec^2(θ)cot^2(θ)-1=cot^2(θ)

解答

证明

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

解答

真

求解步骤

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

调整左侧

sec^{2} (θ) cot^{2} (θ) - 1

用 sin, cos 表示

- 1 + cot^{2} (θ) sec^{2} (θ)

使用基本三角恒等式:

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

= - 1 + (\frac{cos ( θ )}{sin ( θ )})^{2} sec^{2} (θ)

使用基本三角恒等式:

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

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

化简

- 1 + (\frac{cos ( θ )}{sin ( θ )})^{2} (\frac{1}{cos ( θ )})^{2} : \frac{- sin ^{2} ( θ ) + 1}{sin ^{2} ( θ )}

- 1 + (\frac{cos ( θ )}{sin ( θ )})^{2} (\frac{1}{cos ( θ )})^{2}

(\frac{cos ( θ )}{sin ( θ )})^{2} (\frac{1}{cos ( θ )})^{2} = \frac{1}{sin ^{2} ( θ )}

(\frac{cos ( θ )}{sin ( θ )})^{2} (\frac{1}{cos ( θ )})^{2}

(\frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

(\frac{cos ( θ )}{sin ( θ )})^{2}

使用指数法则:

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

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

= (\frac{1}{cos ( θ )})^{2} \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

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

(\frac{1}{cos ( θ )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} \cdot \frac{1}{cos ^{2} ( θ )}

分式相乘:

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

= \frac{cos ^{2} ( θ ) \cdot 1}{sin ^{2} ( θ ) cos ^{2} ( θ )}

约分:

cos^{2} (θ)

= \frac{1}{sin ^{2} ( θ )}

= - 1 + \frac{1}{sin ^{2} ( θ )}

将项转换为分式:

1 = \frac{1 sin ^{2} ( θ )}{sin ^{2} ( θ )}

= - \frac{1 \cdot sin ^{2} ( θ )}{sin ^{2} ( θ )} + \frac{1}{sin ^{2} ( θ )}

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 sin ^{2} ( θ ) + 1}{sin ^{2} ( θ )}

乘以:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

= \frac{- sin ^{2} ( θ ) + 1}{sin ^{2} ( θ )}

= \frac{- sin ^{2} ( θ ) + 1}{sin ^{2} ( θ )}

= \frac{1 - sin ^{2} ( θ )}{sin ^{2} ( θ )}

调整右侧

cot^{2} (θ)

用 sin, cos 表示

cot^{2} (θ)

使用基本三角恒等式:

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

= (\frac{cos ( θ )}{sin ( θ )})^{2}

化简

(\frac{cos ( θ )}{sin ( θ )})^{2} : \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

(\frac{cos ( θ )}{sin ( θ )})^{2}

使用指数法则:

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

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

使用三角恒等式改写

\frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

使用毕达哥拉斯恒等式:

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

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

= \frac{1 - sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{1 - sin ^{2} ( θ )}{sin ^{2} ( θ )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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