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受欢迎的三角函数 >

证明 cos(2a)=(cot(a)-tan(a))/(csc(a)sec(a))

解答

证明

cos (2 a) = \frac{cot ( a ) - tan ( a )}{csc ( a ) sec ( a )}

解答

真

求解步骤

cos (2 a) = \frac{cot ( a ) - tan ( a )}{csc ( a ) sec ( a )}

调整右侧

\frac{cot ( a ) - tan ( a )}{csc ( a ) sec ( a )}

用 sin, cos 表示

\frac{cot ( a ) - tan ( a )}{csc ( a ) sec ( a )}

使用基本三角恒等式:

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

= \frac{\frac{c o s ( a )}{s i n ( a )} - tan ( a )}{csc ( a ) sec ( a )}

使用基本三角恒等式:

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

= \frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{csc ( a ) sec ( a )}

使用基本三角恒等式:

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

= \frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{\frac{1}{s i n ( a )} sec ( a )}

使用基本三角恒等式:

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

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

化简

\frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{\frac{1}{s i n ( a )} \cdot \frac{1}{c o s ( a )}} : cos^{2} (a) - sin^{2} (a)

\frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{\frac{1}{s i n ( a )} \cdot \frac{1}{c o s ( a )}}

乘

\frac{1}{sin ( a )} \cdot \frac{1}{cos ( a )} : \frac{1}{sin ( a ) cos ( a )}

\frac{1}{sin ( a )} \cdot \frac{1}{cos ( a )}

分式相乘:

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

= \frac{1 \cdot 1}{sin ( a ) cos ( a )}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{sin ( a ) cos ( a )}

= \frac{\frac{c o s ( a )}{s i n ( a )} - \frac{s i n ( a )}{c o s ( a )}}{\frac{1}{s i n ( a ) c o s ( a )}}

化简

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

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

sin (a), cos (a)

的最小公倍数:

sin (a) cos (a)

sin (a), cos (a)

最小公倍数 (LCM)

计算出由出现在

sin (a)

或

cos (a)

中的因子组成的表达式

= sin (a) cos (a)

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 (a) cos (a)

对于

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

将分母和分子乘以

cos (a)

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

对于

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

将分母和分子乘以

sin (a)

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

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

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

= \frac{\frac{c o s ^{2} ( a ) - s i n ^{2} ( a )}{s i n ( a ) c o s ( a )}}{\frac{1}{s i n ( a ) c o s ( a )}}

分式相除:

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

= \frac{( cos ^{2} ( a ) - sin ^{2} ( a ) ) sin ( a ) cos ( a )}{sin ( a ) cos ( a ) \cdot 1}

整理后得

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

约分:

sin (a)

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

约分:

cos (a)

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

= cos (2 a)

= cos (2 a)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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