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受欢迎的三角函数 >

证明 (csc^2(t))/(cot(t))=csc(t)sec(t)

解答

证明

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

解答

真

求解步骤

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

调整左侧

\frac{csc ^{2} ( t )}{cot ( t )}

用 sin, cos 表示

\frac{csc ^{2} ( t )}{cot ( t )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

使用分式法则:

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

= \frac{( \frac{1}{s i n ( t )} ) ^{2} sin ( t )}{cos ( t )}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{s i n ^{2} ( t )} sin ( t )}{cos ( t )}

乘

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

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

分式相乘:

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

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

乘以:

1 \cdot sin (t) = sin (t)

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

约分:

sin (t)

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

= \frac{\frac{1}{s i n ( t )}}{cos ( t )}

使用分式法则:

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

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

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

= csc (t) sec (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 (2cos(2x))/(sin(2x))=cot(x)-tan(x)

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

证明 cot^2(t)-cos^2(t)=cot^2(t)cos^2(t)

p ro v e cot^{2} (t) - cos^{2} (t) = cot^{2} (t) cos^{2} (t)

证明 cot(θ)=(cos(θ))/(sin(θ))

p ro v e cot (θ) = \frac{cos ( θ )}{sin ( θ )}

证明 (1-cot(x))^2=csc^2(x)-2cot(x)

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

证明 1/(cot(x)(1-cos(2x)))=csc(2x)

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