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受欢迎的三角函数 >

证明 1/(csc(x)-1)+1/(csc(x)+1)=2tan(x)sec(x)

解答

证明

\frac{1}{csc ( x ) - 1} + \frac{1}{csc ( x ) + 1} = 2 tan (x) sec (x)

解答

真

求解步骤

\frac{1}{csc ( x ) - 1} + \frac{1}{csc ( x ) + 1} = 2 tan (x) sec (x)

调整左侧

\frac{1}{csc ( x ) - 1} + \frac{1}{csc ( x ) + 1}

用 sin, cos 表示

\frac{1}{- 1 + csc ( x )} + \frac{1}{1 + csc ( x )}

使用基本三角恒等式:

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

= \frac{1}{- 1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + \frac{1}{s i n ( x )}}

化简

\frac{1}{- 1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + \frac{1}{s i n ( x )}} : \frac{2 sin ( x )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

\frac{1}{- 1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + \frac{1}{s i n ( x )}}

\frac{1}{- 1 + \frac{1}{s i n ( x )}} = \frac{sin ( x )}{- sin ( x ) + 1}

\frac{1}{- 1 + \frac{1}{s i n ( x )}}

化简

- 1 + \frac{1}{sin ( x )} : \frac{- sin ( x ) + 1}{sin ( x )}

- 1 + \frac{1}{sin ( x )}

将项转换为分式:

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

= - \frac{1 \cdot sin ( x )}{sin ( x )} + \frac{1}{sin ( 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{- 1 \cdot sin ( x ) + 1}{sin ( x )}

乘以:

1 \cdot sin (x) = sin (x)

= \frac{- sin ( x ) + 1}{sin ( x )}

= \frac{1}{\frac{- s i n ( x ) + 1}{s i n ( x )}}

使用分式法则:

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

= \frac{sin ( x )}{- sin ( x ) + 1}

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

\frac{1}{1 + \frac{1}{s i n ( x )}}

化简

1 + \frac{1}{sin ( x )} : \frac{sin ( x ) + 1}{sin ( x )}

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

将项转换为分式:

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

= \frac{1 \cdot sin ( x )}{sin ( x )} + \frac{1}{sin ( 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{1 \cdot sin ( x ) + 1}{sin ( x )}

乘以:

1 \cdot sin (x) = sin (x)

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

= \frac{1}{\frac{s i n ( x ) + 1}{s i n ( x )}}

使用分式法则:

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

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

= \frac{sin ( x )}{- sin ( x ) + 1} + \frac{sin ( x )}{sin ( x ) + 1}

- sin (x) + 1, sin (x) + 1

的最小公倍数:

(sin (x) + 1) (- sin (x) + 1)

- sin (x) + 1, sin (x) + 1

最小公倍数 (LCM)

计算出由出现在

- sin (x) + 1

或

sin (x) + 1

中的因子组成的表达式

= (sin (x) + 1) (- sin (x) + 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 (x) + 1) (- sin (x) + 1)

对于

\frac{sin ( x )}{- sin ( x ) + 1} :

将分母和分子乘以

sin (x) + 1

\frac{sin ( x )}{- sin ( x ) + 1} = \frac{sin ( x ) ( sin ( x ) + 1 )}{( - sin ( x ) + 1 ) ( sin ( x ) + 1 )}

对于

\frac{sin ( x )}{sin ( x ) + 1} :

将分母和分子乘以

- sin (x) + 1

\frac{sin ( x )}{sin ( x ) + 1} = \frac{sin ( x ) ( - sin ( x ) + 1 )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

= \frac{sin ( x ) ( sin ( x ) + 1 )}{( - sin ( x ) + 1 ) ( sin ( x ) + 1 )} + \frac{sin ( x ) ( - sin ( x ) + 1 )}{( sin ( x ) + 1 ) ( - sin ( x ) + 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{sin ( x ) ( sin ( x ) + 1 ) + sin ( x ) ( - sin ( x ) + 1 )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

乘开

sin (x) (sin (x) + 1) + sin (x) (- sin (x) + 1) : 2 sin (x)

sin (x) (sin (x) + 1) + sin (x) (- sin (x) + 1)

乘开

sin (x) (sin (x) + 1) : sin^{2} (x) + sin (x)

sin (x) (sin (x) + 1)

使用分配律:

a (b + c) = ab + a c

a = sin (x), b = sin (x), c = 1

= sin (x) sin (x) + sin (x) \cdot 1

= sin (x) sin (x) + 1 \cdot sin (x)

化简

sin (x) sin (x) + 1 \cdot sin (x) : sin^{2} (x) + sin (x)

sin (x) sin (x) + 1 \cdot sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= sin (x)

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

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

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

乘开

sin (x) (- sin (x) + 1) : - sin^{2} (x) + sin (x)

sin (x) (- sin (x) + 1)

使用分配律:

a (b + c) = ab + a c

a = sin (x), b = - sin (x), c = 1

= sin (x) (- sin (x)) + sin (x) \cdot 1

使用加减运算法则

+ (- a) = - a

= - sin (x) sin (x) + 1 \cdot sin (x)

化简

- sin (x) sin (x) + 1 \cdot sin (x) : - sin^{2} (x) + sin (x)

- sin (x) sin (x) + 1 \cdot sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= sin (x)

= - sin^{2} (x) + sin (x)

= - sin^{2} (x) + sin (x)

= sin^{2} (x) + sin (x) - sin^{2} (x) + sin (x)

化简

sin^{2} (x) + sin (x) - sin^{2} (x) + sin (x) : 2 sin (x)

sin^{2} (x) + sin (x) - sin^{2} (x) + sin (x)

同类项相加:

sin^{2} (x) - sin^{2} (x) = 0

= sin (x) + sin (x)

同类项相加:

sin (x) + sin (x) = 2 sin (x)

= 2 sin (x)

= 2 sin (x)

= \frac{2 sin ( x )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

= \frac{2 sin ( x )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

= \frac{2 sin ( x )}{( 1 + sin ( x ) ) ( 1 - sin ( x ) )}

使用三角恒等式改写

\frac{2 sin ( x )}{( 1 + sin ( x ) ) ( 1 - sin ( x ) )}

乘开

(1 + sin (x)) (1 - sin (x)) : 1 - sin^{2} (x)

(1 + sin (x)) (1 - sin (x))

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (x)

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

= \frac{2 sin ( x )}{1 - sin ^{2} ( x )}

使用毕达哥拉斯恒等式:

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

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

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

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

调整右侧

2 tan (x) sec (x)

用 sin, cos 表示

2 sec (x) tan (x)

使用基本三角恒等式:

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

= 2 \cdot \frac{1}{cos ( x )} tan (x)

使用基本三角恒等式:

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

= 2 \cdot \frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )}

化简

2 \cdot \frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ^{2} ( x )}

2 \cdot \frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )}

分式相乘:

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

= \frac{1 \cdot sin ( x ) \cdot 2}{cos ( x ) cos ( x )}

数字相乘:

1 \cdot 2 = 2

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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 (1+cos(t))/(1-cos(t))=(csc(t)+cot(t))^2

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

证明 sec^2(α)+csc^2(α)= 1/(sin^2(α)cos^2(α))

p ro v e sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

证明 sin^2(A)+cos^2(A)=1

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

证明 csc(2x)+cot(2x)=cot(x)

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

证明 cos^2(t)=(1+cos(2t))/2

p ro v e cos^{2} (t) = \frac{1 + cos ( 2 t )}{2}