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受欢迎的三角函数 >

证明 1/(tan(x))+tan(x)=sec(x)csc(x)

解答

证明

\frac{1}{tan ( x )} + tan (x) = sec (x) csc (x)

解答

真

求解步骤

\frac{1}{tan ( x )} + tan (x) = sec (x) csc (x)

调整左侧

\frac{1}{tan ( x )} + tan (x)

用 sin, cos 表示

\frac{1}{tan ( x )} + tan (x)

使用基本三角恒等式:

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

= \frac{1}{\frac{s i n ( x )}{c o s ( x )}} + \frac{sin ( x )}{cos ( x )}

化简

\frac{1}{\frac{s i n ( x )}{c o s ( x )}} + \frac{sin ( x )}{cos ( x )} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{1}{\frac{s i n ( x )}{c o s ( x )}} + \frac{sin ( x )}{cos ( x )}

使用分式法则:

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

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 (x) cos (x)

对于

\frac{cos ( x )}{sin ( x )} :

将分母和分子乘以

cos (x)

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

对于

\frac{sin ( x )}{cos ( x )} :

将分母和分子乘以

sin (x)

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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 ( x ) \frac{1}{c s c ( x )}}

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( x )} \cdot \frac{1}{c s c ( x )}}

化简

\frac{1}{\frac{1}{s e c ( x )} \cdot \frac{1}{c s c ( x )}}

乘

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )} : \frac{1}{sec ( x ) csc ( x )}

\frac{1}{sec ( x )} \cdot \frac{1}{csc ( x )}

分式相乘:

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

= \frac{1 \cdot 1}{sec ( x ) csc ( x )}

数字相乘:

1 \cdot 1 = 1

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

= \frac{1}{\frac{1}{s e c ( x ) c s c ( x )}}

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= sec (x) csc (x)

sec (x) csc (x)

sec (x) csc (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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p ro v e tan (θ) + cot (θ) = 2 csc (2 θ)

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p ro v e csc^{4} (x) - cot^{4} (x) = 2 cot^{2} (x) + 1

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p ro v e cos (- x) - sin (- x) = cos (x) + sin (x)

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p ro v e cos (θ) sec (θ) = 1

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