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受欢迎的三角函数 >

证明 2tan(x)sec(x)csc(x)=2+2tan^2(x)

解答

证明

2 tan (x) sec (x) csc (x) = 2 + 2 tan^{2} (x)

解答

真

求解步骤

2 tan (x) sec (x) csc (x) = 2 + 2 tan^{2} (x)

调整左侧

2 tan (x) sec (x) csc (x)

用 sin, cos 表示

2 csc (x) sec (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

约分:

sin (x)

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

数字相乘:

1 \cdot 1 \cdot 2 = 2

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

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

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

调整右侧

2 + 2 tan^{2} (x)

分解

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

2 + 2 tan^{2} (x)

改写为

= 2 \cdot 1 + 2 tan^{2} (x)

因式分解出通项

2

= 2 (1 + tan^{2} (x))

= (1 + tan^{2} (x)) \cdot 2

用 sin, cos 表示

(1 + tan^{2} (x)) \cdot 2

使用基本三角恒等式:

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

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

化简

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

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

使用指数法则:

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

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

化简

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

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

将项转换为分式:

1 = \frac{1 cos ^{2} ( x )}{cos ^{2} ( x )}

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

乘以:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

化简

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

= \frac{2}{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

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