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受欢迎的三角函数 >

证明 (csc(B))/(tan(B)+cot(B))=cos(B)

解答

证明

\frac{csc ( B )}{tan ( B ) + cot ( B )} = cos (B)

解答

真

求解步骤

\frac{csc ( B )}{tan ( B ) + cot ( B )} = cos (B)

调整左侧

\frac{csc ( B )}{tan ( B ) + cot ( B )}

用 sin, cos 表示

\frac{csc ( B )}{cot ( B ) + tan ( B )}

使用基本三角恒等式:

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

= \frac{\frac{1}{s i n ( B )}}{cot ( B ) + tan ( B )}

使用基本三角恒等式:

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

= \frac{\frac{1}{s i n ( B )}}{\frac{c o s ( B )}{s i n ( B )} + tan ( B )}

使用基本三角恒等式:

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

= \frac{\frac{1}{s i n ( B )}}{\frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}

化简

\frac{\frac{1}{s i n ( B )}}{\frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}} : \frac{cos ( B )}{cos ^{2} ( B ) + sin ^{2} ( B )}

\frac{\frac{1}{s i n ( B )}}{\frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )}}

使用分式法则:

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

= \frac{1}{sin ( B ) ( \frac{c o s ( B )}{s i n ( B )} + \frac{s i n ( B )}{c o s ( B )} )}

化简

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

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

sin (B), cos (B)

的最小公倍数:

sin (B) cos (B)

sin (B), cos (B)

最小公倍数 (LCM)

计算出由出现在

sin (B)

或

cos (B)

中的因子组成的表达式

= sin (B) cos (B)

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 (B) cos (B)

对于

\frac{cos ( B )}{sin ( B )} :

将分母和分子乘以

cos (B)

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

对于

\frac{sin ( B )}{cos ( B )} :

将分母和分子乘以

sin (B)

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

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

= \frac{1}{\frac{c o s ^{2} ( B ) + s i n ^{2} ( B )}{s i n ( B ) c o s ( B )} sin ( B )}

乘

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

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

分式相乘:

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

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

约分:

sin (B)

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

= \frac{1}{\frac{c o s ^{2} ( B ) + s i n ^{2} ( B )}{c o s ( B )}}

使用分式法则:

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

使用法则

\frac{a}{1} = a

= cos (B)

= cos (B)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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