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受欢迎的三角函数 >

证明 sin(b)+cos(b)cot(b)=csc(b)

解答

证明

sin (b) + cos (b) cot (b) = csc (b)

解答

真

求解步骤

sin (b) + cos (b) cot (b) = csc (b)

调整左侧

sin (b) + cos (b) cot (b)

用 sin, cos 表示

sin (b) + cos (b) cot (b)

使用基本三角恒等式:

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

= sin (b) + cos (b) \frac{cos ( b )}{sin ( b )}

化简

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

sin (b) + cos (b) \frac{cos ( b )}{sin ( b )}

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

cos (b) \frac{cos ( b )}{sin ( b )}

分式相乘:

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

= \frac{cos ( b ) cos ( b )}{sin ( b )}

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

cos (b) cos (b)

使用指数法则:

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

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

= cos^{1 + 1} (b)

数字相加:

1 + 1 = 2

= cos^{2} (b)

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

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

将项转换为分式:

sin (b) = \frac{sin ( b ) sin ( b )}{sin ( b )}

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

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 ( b ) sin ( b ) + cos ^{2} ( b )}{sin ( b )}

sin (b) sin (b) + cos^{2} (b) = sin^{2} (b) + cos^{2} (b)

sin (b) sin (b) + cos^{2} (b)

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

sin (b) sin (b)

使用指数法则:

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

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

= sin^{1 + 1} (b)

数字相加:

1 + 1 = 2

= sin^{2} (b)

= sin^{2} (b) + cos^{2} (b)

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

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

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

使用三角恒等式改写

\frac{cos ^{2} ( b ) + sin ^{2} ( b )}{sin ( b )}

使用毕达哥拉斯恒等式:

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

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

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

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 ( b )}}

化简

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

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= csc (b)

csc (b)

csc (b)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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