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受欢迎的三角函数 >

证明 (sin(x))/(1-sin(x))+(sin(x))/(1+sin(x))=(2tan(x))/(cos(x))

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

化简

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

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

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

的最小公倍数:

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

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

最小公倍数 (LCM)

计算出由出现在

1 + sin (x)

或

1 - sin (x)

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 )}{1 + sin ( x )} :

将分母和分子乘以

- sin (x) + 1

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

对于

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

将分母和分子乘以

sin (x) + 1

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

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

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 (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^{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 )}{( 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 )}

使用三角恒等式改写

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

使用基本三角恒等式:

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

\frac{2 tan ( x )}{cos ( x )}

\frac{2 tan ( x )}{cos ( x )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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