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受欢迎的三角函数 >

证明 cot^2(y)sec^2(y)=1+cot^2(y)

解答

证明

cot^{2} (y) sec^{2} (y) = 1 + cot^{2} (y)

解答

真

求解步骤

cot^{2} (y) sec^{2} (y) = 1 + cot^{2} (y)

调整左侧

cot^{2} (y) sec^{2} (y)

用 sin, cos 表示

cot^{2} (y) sec^{2} (y)

使用基本三角恒等式:

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

= (\frac{cos ( y )}{sin ( y )})^{2} sec^{2} (y)

使用基本三角恒等式:

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

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

化简

(\frac{cos ( y )}{sin ( y )})^{2} (\frac{1}{cos ( y )})^{2} : \frac{1}{sin ^{2} ( y )}

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

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

(\frac{cos ( y )}{sin ( y )})^{2}

使用指数法则:

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

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

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

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

分式相乘:

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

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

约分:

cos^{2} (y)

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

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

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

调整右侧

1 + cot^{2} (y)

用 sin, cos 表示

1 + cot^{2} (y)

使用基本三角恒等式:

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

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

化简

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

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

使用指数法则:

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

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

将项转换为分式:

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

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

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} ( y ) + cos ^{2} ( y )}{sin ^{2} ( y )}

乘以:

1 \cdot sin^{2} (y) = sin^{2} (y)

= \frac{sin ^{2} ( y ) + cos ^{2} ( y )}{sin ^{2} ( y )}

= \frac{sin ^{2} ( y ) + cos ^{2} ( y )}{sin ^{2} ( y )}

= \frac{cos ^{2} ( y ) + sin ^{2} ( y )}{sin ^{2} ( y )}

使用三角恒等式改写

\frac{cos ^{2} ( y ) + sin ^{2} ( y )}{sin ^{2} ( y )}

使用毕达哥拉斯恒等式:

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

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

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

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(θ)-cot(θ)*cos(θ)=sin(θ)

p ro v e csc (θ) - cot (θ) \cdot cos (θ) = sin (θ)

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

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

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p ro v e sin (a + b) - sin (a - b) = 2 cos (a) sin (b)

证明 sin(2pi-x)=-sin(x)

p ro v e sin (2 π - x) = - sin (x)