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受欢迎的三角函数 >

证明 cot(t)+(sin(t))/(1+cos(t))=csc(t)

解答

证明

cot (t) + \frac{sin ( t )}{1 + cos ( t )} = csc (t)

解答

真

求解步骤

cot (t) + \frac{sin ( t )}{1 + cos ( t )} = csc (t)

调整左侧

cot (t) + \frac{sin ( t )}{1 + cos ( t )}

用 sin, cos 表示

cot (t) + \frac{sin ( t )}{1 + cos ( t )}

使用基本三角恒等式:

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

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

化简

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

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

sin (t), 1 + cos (t)

的最小公倍数:

sin (t) (cos (t) + 1)

sin (t), 1 + cos (t)

最小公倍数 (LCM)

计算出由出现在

sin (t)

或

1 + cos (t)

中的因子组成的表达式

= sin (t) (cos (t) + 1)

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) + 1)

对于

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

将分母和分子乘以

cos (t) + 1

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

对于

\frac{sin ( t )}{1 + cos ( t )} :

将分母和分子乘以

sin (t)

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

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

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 ( t ) ( cos ( t ) + 1 ) + sin ^{2} ( t )}{sin ( t ) ( cos ( t ) + 1 )}

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

1 - cos^{2} (t) + (1 + cos (t)) cos (t) : cos (t) + 1

1 - cos^{2} (t) + (1 + cos (t)) cos (t)

= 1 - cos^{2} (t) + cos (t) (1 + cos (t))

乘开

cos (t) (1 + cos (t)) : cos (t) + cos^{2} (t)

cos (t) (1 + cos (t))

使用分配律:

a (b + c) = ab + a c

a = cos (t), b = 1, c = cos (t)

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

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

化简

1 \cdot cos (t) + cos (t) cos (t) : cos (t) + cos^{2} (t)

1 \cdot cos (t) + cos (t) cos (t)

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

1 \cdot cos (t)

乘以:

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

= cos (t)

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

cos (t) cos (t)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (t) cos (t) = cos^{1 + 1} (t)

= cos^{1 + 1} (t)

数字相加:

1 + 1 = 2

= cos^{2} (t)

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

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

= 1 - cos^{2} (t) + cos (t) + cos^{2} (t)

化简

1 - cos^{2} (t) + cos (t) + cos^{2} (t) : cos (t) + 1

1 - cos^{2} (t) + cos (t) + cos^{2} (t)

对同类项分组

= - cos^{2} (t) + cos (t) + cos^{2} (t) + 1

同类项相加:

- cos^{2} (t) + cos^{2} (t) = 0

= cos (t) + 1

= cos (t) + 1

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

约分:

cos (t) + 1

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

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

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

化简

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

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= csc (t)

csc (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 csc^2(x)*tan^2(x)-1=tan^2(x)

p ro v e csc^{2} (x) \cdot tan^{2} (x) - 1 = tan^{2} (x)

证明 sec^2(x)-cot^2(pi/2-x)=1

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

证明 sin(2x)=-2sin(-x)cos(-x)

p ro v e sin (2 x) = - 2 sin (- x) cos (- x)

证明 sin(θ)csc(θ)-sin^2(θ)=cos^2(θ)

p ro v e sin (θ) csc (θ) - sin^{2} (θ) = cos^{2} (θ)

证明 cos^2(x)csc(x)sec(x)=cot(x)

p ro v e cos^{2} (x) csc (x) sec (x) = cot (x)