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受欢迎的三角函数 >

证明 csc(x)*tan(x)+sec(x)=2sec(x)

解答

证明

csc (x) \cdot tan (x) + sec (x) = 2 sec (x)

解答

真

求解步骤

csc (x) tan (x) + sec (x) = 2 sec (x)

调整左侧

csc (x) tan (x) + sec (x)

用 sin, cos 表示

sec (x) + csc (x) tan (x)

使用基本三角恒等式:

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

= \frac{1}{cos ( x )} + csc (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

约分:

sin (x)

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

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

使用法则

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

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

数字相加:

1 + 1 = 2

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{2}{\frac{1}{s e c ( x )}}

化简

\frac{2}{\frac{1}{s e c ( x )}}

使用分式法则:

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

= \frac{2 sec ( x )}{1}

使用法则

\frac{a}{1} = a

= 2 sec (x)

2 sec (x)

2 sec (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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