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受欢迎的三角函数 >

证明 1/2 (cot(θ)-tan(θ))=cot(2θ)

解答

证明

\frac{1}{2} (cot (θ) - tan (θ)) = cot (2 θ)

解答

真

求解步骤

\frac{1}{2} (cot (θ) - tan (θ)) = cot (2 θ)

调整左侧

\frac{1}{2} (cot (θ) - tan (θ))

用 sin, cos 表示

(cot (θ) - tan (θ)) \frac{1}{2}

使用基本三角恒等式:

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

= (\frac{cos ( θ )}{sin ( θ )} - tan (θ)) \frac{1}{2}

使用基本三角恒等式:

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

= (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}) \frac{1}{2}

化简

(\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}) \frac{1}{2} : \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{2 sin ( θ ) cos ( θ )}

(\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}) \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot ( \frac{c o s ( θ )}{s i n ( θ )} - \frac{s i n ( θ )}{c o s ( θ )} )}{2}

1 \cdot (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}) = \frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}

1 \cdot (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )})

乘以:

1 \cdot (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}) = (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )})

= (\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )})

去除括号:

(a) = a

= \frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{s i n ( θ )}{c o s ( θ )}}{2}

化简

\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )} : \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{sin ( θ ) cos ( θ )}

\frac{cos ( θ )}{sin ( θ )} - \frac{sin ( θ )}{cos ( θ )}

sin (θ), cos (θ)

的最小公倍数:

sin (θ) cos (θ)

sin (θ), cos (θ)

最小公倍数 (LCM)

计算出由出现在

sin (θ)

或

cos (θ)

中的因子组成的表达式

= sin (θ) cos (θ)

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 (θ) cos (θ)

对于

\frac{cos ( θ )}{sin ( θ )} :

将分母和分子乘以

cos (θ)

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

对于

\frac{sin ( θ )}{cos ( θ )} :

将分母和分子乘以

sin (θ)

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

= \frac{cos ^{2} ( θ )}{sin ( θ ) cos ( θ )} - \frac{sin ^{2} ( θ )}{sin ( θ ) cos ( θ )}

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} ( θ ) - sin ^{2} ( θ )}{sin ( θ ) cos ( θ )}

= \frac{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{s i n ( θ ) c o s ( θ )}}{2}

使用分式法则:

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

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{sin ( θ ) cos ( θ ) \cdot 2}

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{2 sin ( θ ) cos ( θ )}

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{2 cos ( θ ) sin ( θ )}

使用三角恒等式改写

\frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{2 cos ( θ ) sin ( θ )}

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{sin ( 2 θ )}

使用倍角公式:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= \frac{cos ( 2 θ )}{sin ( 2 θ )}

= \frac{cos ( 2 θ )}{sin ( 2 θ )}

使用基本三角恒等式:

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

= cot (2 θ)

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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