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受欢迎的三角函数 >

证明 7/(tan(x))+7/(cot(x))=7tan(x)+7cot(x)

解答

证明

\frac{7}{tan ( x )} + \frac{7}{cot ( x )} = 7 tan (x) + 7 cot (x)

解答

真

求解步骤

\frac{7}{tan ( x )} + \frac{7}{cot ( x )} = 7 tan (x) + 7 cot (x)

调整左侧

\frac{7}{tan ( x )} + \frac{7}{cot ( x )}

用 sin, cos 表示

\frac{7}{cot ( x )} + \frac{7}{tan ( x )}

使用基本三角恒等式:

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

= \frac{7}{\frac{c o s ( x )}{s i n ( x )}} + \frac{7}{tan ( x )}

使用基本三角恒等式:

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

= \frac{7}{\frac{c o s ( x )}{s i n ( x )}} + \frac{7}{\frac{s i n ( x )}{c o s ( x )}}

化简

\frac{7}{\frac{c o s ( x )}{s i n ( x )}} + \frac{7}{\frac{s i n ( x )}{c o s ( x )}} : \frac{7 sin ^{2} ( x ) + 7 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{7}{\frac{c o s ( x )}{s i n ( x )}} + \frac{7}{\frac{s i n ( x )}{c o s ( x )}}

使用分式法则:

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

= \frac{7 sin ( x )}{cos ( x )} + \frac{7}{\frac{s i n ( x )}{c o s ( x )}}

使用分式法则:

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

= \frac{7 sin ( x )}{cos ( x )} + \frac{7 cos ( x )}{sin ( x )}

cos (x), sin (x)

的最小公倍数:

cos (x) sin (x)

cos (x), sin (x)

最小公倍数 (LCM)

计算出由出现在

cos (x)

或

sin (x)

中的因子组成的表达式

= cos (x) sin (x)

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

cos (x) sin (x)

对于

\frac{7 sin ( x )}{cos ( x )} :

将分母和分子乘以

sin (x)

\frac{7 sin ( x )}{cos ( x )} = \frac{7 sin ( x ) sin ( x )}{cos ( x ) sin ( x )} = \frac{7 sin ^{2} ( x )}{cos ( x ) sin ( x )}

对于

\frac{7 cos ( x )}{sin ( x )} :

将分母和分子乘以

cos (x)

\frac{7 cos ( x )}{sin ( x )} = \frac{7 cos ( x ) cos ( x )}{sin ( x ) cos ( x )} = \frac{7 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{7 sin ^{2} ( x )}{cos ( x ) sin ( x )} + \frac{7 cos ^{2} ( x )}{cos ( x ) sin ( 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{7 sin ^{2} ( x ) + 7 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{7 sin ^{2} ( x ) + 7 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{7 cos ^{2} ( x ) + 7 sin ^{2} ( x )}{cos ( x ) sin ( x )}

调整右侧

7 tan (x) + 7 cot (x)

用 sin, cos 表示

7 cot (x) + 7 tan (x)

使用基本三角恒等式:

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

= 7 \cdot \frac{cos ( x )}{sin ( x )} + 7 tan (x)

使用基本三角恒等式:

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

= 7 \cdot \frac{cos ( x )}{sin ( x )} + 7 \cdot \frac{sin ( x )}{cos ( x )}

化简

7 \cdot \frac{cos ( x )}{sin ( x )} + 7 \cdot \frac{sin ( x )}{cos ( x )} : \frac{7 cos ^{2} ( x ) + 7 sin ^{2} ( x )}{sin ( x ) cos ( x )}

7 \cdot \frac{cos ( x )}{sin ( x )} + 7 \cdot \frac{sin ( x )}{cos ( x )}

乘

7 \cdot \frac{cos ( x )}{sin ( x )} : \frac{7 cos ( x )}{sin ( x )}

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

分式相乘:

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

= \frac{cos ( x ) \cdot 7}{sin ( x )}

= \frac{7 cos ( x )}{sin ( x )} + 7 \cdot \frac{sin ( x )}{cos ( x )}

乘

7 \cdot \frac{sin ( x )}{cos ( x )} : \frac{7 sin ( x )}{cos ( x )}

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

分式相乘:

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

= \frac{sin ( x ) \cdot 7}{cos ( x )}

= \frac{7 cos ( x )}{sin ( x )} + \frac{7 sin ( x )}{cos ( x )}

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

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) cos (x)

对于

\frac{cos ( x ) \cdot 7}{sin ( x )} :

将分母和分子乘以

cos (x)

\frac{cos ( x ) \cdot 7}{sin ( x )} = \frac{cos ( x ) \cdot 7 cos ( x )}{sin ( x ) cos ( x )} = \frac{7 cos ^{2} ( x )}{sin ( x ) cos ( x )}

对于

\frac{sin ( x ) \cdot 7}{cos ( x )} :

将分母和分子乘以

sin (x)

\frac{sin ( x ) \cdot 7}{cos ( x )} = \frac{sin ( x ) \cdot 7 sin ( x )}{cos ( x ) sin ( x )} = \frac{7 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{7 cos ^{2} ( x )}{sin ( x ) cos ( x )} + \frac{7 sin ^{2} ( x )}{sin ( x ) cos ( 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{7 cos ^{2} ( x ) + 7 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{7 cos ^{2} ( x ) + 7 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{7 cos ^{2} ( x ) + 7 sin ^{2} ( x )}{cos ( x ) sin ( x )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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