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受欢迎的三角函数 >

证明 sec(x)=sin(x)(cot(x)+tan(x))

解答

证明

sec (x) = sin (x) (cot (x) + tan (x))

解答

真

求解步骤

sec (x) = sin (x) (cot (x) + tan (x))

调整右侧

sin (x) (cot (x) + tan (x))

用 sin, cos 表示

(cot (x) + tan (x)) sin (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

分式相乘:

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

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

约分:

sin (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{s e c ( x )}}

化简

\frac{1}{\frac{1}{s e c ( x )}}

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 (tan^2(θ)+1)/(tan^2(θ))=csc^2(θ)

p ro v e \frac{tan ^{2} ( θ ) + 1}{tan ^{2} ( θ )} = csc^{2} (θ)

证明 sec^2(x)csc^2(x)-sec^2(x)=csc^2(x)

p ro v e sec^{2} (x) csc^{2} (x) - sec^{2} (x) = csc^{2} (x)

证明 (sin(θ)+sin(3θ))/(2cos(θ))=sin(2θ)

p ro v e \frac{sin ( θ ) + sin ( 3 θ )}{2 cos ( θ )} = sin (2 θ)

证明 csc(θ)-csc(θ)cos^2(θ)=sin(θ)

p ro v e csc (θ) - csc (θ) cos^{2} (θ) = sin (θ)

证明 cot(2β)=(cot^2(β)-1)/(2cot(β))

p ro v e cot (2 β) = \frac{cot ^{2} ( β ) - 1}{2 cot ( β )}