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受欢迎的三角函数 >

证明 tan^2(θ)csc^2(θ)=sec^2(θ)

解答

证明

tan^{2} (θ) csc^{2} (θ) = sec^{2} (θ)

解答

真

求解步骤

tan^{2} (θ) csc^{2} (θ) = sec^{2} (θ)

调整左侧

tan^{2} (θ) csc^{2} (θ)

用 sin, cos 表示

csc^{2} (θ) tan^{2} (θ)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

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

使用基本三角恒等式:

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

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

化简

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

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

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

(\frac{1}{sin ( θ )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

使用指数法则:

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

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

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

分式相乘:

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

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

约分:

sin^{2} (θ)

= \frac{1}{cos ^{2} ( θ )}

= \frac{1}{cos ^{2} ( θ )}

= \frac{1}{cos ^{2} ( θ )}

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{( \frac{1}{s e c ( θ )} ) ^{2}}

化简

\frac{1}{( \frac{1}{s e c ( θ )} ) ^{2}}

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

(\frac{1}{sec ( θ )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sec ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( θ )}

= \frac{1}{\frac{1}{s e c ^{2} ( θ )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ^{2} ( θ )}{1}

使用法则

\frac{a}{1} = a

= sec^{2} (θ)

sec^{2} (θ)

sec^{2} (θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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