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受欢迎的三角函数 >

证明 (sec^2(t))/(tan(t))=cot(t)+tan(t)

解答

证明

\frac{sec ^{2} ( t )}{tan ( t )} = cot (t) + tan (t)

解答

真

求解步骤

\frac{sec ^{2} ( t )}{tan ( t )} = cot (t) + tan (t)

调整左侧

\frac{sec ^{2} ( t )}{tan ( t )}

用 sin, cos 表示

\frac{sec ^{2} ( t )}{tan ( t )}

使用基本三角恒等式:

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

= \frac{( \frac{1}{c o s ( t )} ) ^{2}}{tan ( t )}

使用基本三角恒等式:

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

= \frac{( \frac{1}{c o s ( t )} ) ^{2}}{\frac{s i n ( t )}{c o s ( t )}}

化简

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

\frac{( \frac{1}{c o s ( t )} ) ^{2}}{\frac{s i n ( t )}{c o s ( t )}}

使用分式法则:

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

= \frac{( \frac{1}{c o s ( t )} ) ^{2} cos ( t )}{sin ( t )}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{c o s ^{2} ( t )} cos ( t )}{sin ( t )}

乘

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

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

分式相乘:

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

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

乘以:

1 \cdot cos (t) = cos (t)

= \frac{cos ( t )}{cos ^{2} ( t )}

约分:

cos (t)

= \frac{1}{cos ( t )}

= \frac{\frac{1}{c o s ( t )}}{sin ( t )}

使用分式法则:

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

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

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

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

调整右侧

cot (t) + tan (t)

用 sin, cos 表示

cot (t) + tan (t)

使用基本三角恒等式:

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

= \frac{cos ( t )}{sin ( t )} + tan (t)

使用基本三角恒等式:

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

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

化简

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

\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

sin (t), cos (t)

的最小公倍数:

sin (t) cos (t)

sin (t), cos (t)

最小公倍数 (LCM)

计算出由出现在

sin (t)

或

cos (t)

中的因子组成的表达式

= sin (t) cos (t)

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 (t) cos (t)

对于

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

将分母和分子乘以

cos (t)

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

对于

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

将分母和分子乘以

sin (t)

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

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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 csc (\frac{π}{2} - x) = sec (x)

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p ro v e cos (x) cot (x) = \frac{1 - sin ^{2} ( x )}{sin ( x )}

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p ro v e \frac{1}{cos ( x )} - cos (x) = sin (x) tan (x)

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p ro v e sec (u) cos (u) - sin^{2} (u) = cos^{2} (u)