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受欢迎的三角函数 >

证明 sec^2(a)+csc^2(a)=sec^2(a)csc^2(a)

解答

证明

sec^{2} (a) + csc^{2} (a) = sec^{2} (a) csc^{2} (a)

解答

真

求解步骤

sec^{2} (a) + csc^{2} (a) = sec^{2} (a) csc^{2} (a)

调整左侧

sec^{2} (a) + csc^{2} (a)

用 sin, cos 表示

csc^{2} (a) + sec^{2} (a)

使用基本三角恒等式:

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

= (\frac{1}{sin ( a )})^{2} + sec^{2} (a)

使用基本三角恒等式:

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

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

化简

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

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

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

(\frac{1}{sin ( a )})^{2}

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

sin^{2} (a), cos^{2} (a)

的最小公倍数:

sin^{2} (a) cos^{2} (a)

sin^{2} (a), cos^{2} (a)

最小公倍数 (LCM)

计算出由出现在

sin^{2} (a)

或

cos^{2} (a)

中的因子组成的表达式

= sin^{2} (a) cos^{2} (a)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin^{2} (a) cos^{2} (a)

对于

\frac{1}{sin ^{2} ( a )} :

将分母和分子乘以

cos^{2} (a)

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

对于

\frac{1}{cos ^{2} ( a )} :

将分母和分子乘以

sin^{2} (a)

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

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{cos ^{2} ( a ) ( \frac{1}{c s c ( a )} ) ^{2}}

使用基本三角恒等式:

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

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

化简

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

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

(\frac{1}{sec ( a )})^{2}

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

(\frac{1}{csc ( a )})^{2} = \frac{1}{csc ^{2} ( a )}

(\frac{1}{csc ( a )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{csc ^{2} ( a )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( a )}

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

乘

\frac{1}{sec ^{2} ( a )} \cdot \frac{1}{csc ^{2} ( a )} : \frac{1}{sec ^{2} ( a ) csc ^{2} ( a )}

\frac{1}{sec ^{2} ( a )} \cdot \frac{1}{csc ^{2} ( a )}

分式相乘:

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

= \frac{1 \cdot 1}{sec ^{2} ( a ) csc ^{2} ( a )}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{sec ^{2} ( a ) csc ^{2} ( a )}

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

使用分式法则:

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

= \frac{sec ^{2} ( a ) csc ^{2} ( a )}{1}

使用法则

\frac{a}{1} = a

= sec^{2} (a) csc^{2} (a)

sec^{2} (a) csc^{2} (a)

sec^{2} (a) csc^{2} (a)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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