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受欢迎的三角函数 >

证明 (sec^2(t))/(tan(t))=sec(t)csc(t)

解答

证明

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

解答

真

求解步骤

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

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

使用基本三角恒等式:

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

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

化简

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

乘

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

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

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= sec (t) csc (t)

sec (t) csc (t)

sec (t) csc (t)

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 cos(x)(tan^2(x)+1)=sec(x)

p ro v e cos (x) (tan^{2} (x) + 1) = sec (x)

证明 csc^2(θ)-cot^2(θ)=1

p ro v e csc^{2} (θ) - cot^{2} (θ) = 1

证明 (sec^2(x)-1)csc^2(x)=sec^2(x)

p ro v e (sec^{2} (x) - 1) csc^{2} (x) = sec^{2} (x)

证明 (1+sin(a))(1-sin(a))=cos^2(a)

p ro v e (1 + sin (a)) (1 - sin (a)) = cos^{2} (a)

证明 sin^3(θ)= 3/4 sin(θ)-1/4 sin(3θ)

p ro v e sin^{3} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)