zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

证明 cos(θ)+cos(θ)tan^2(θ)=sec(θ)

解答

证明

cos (θ) + cos (θ) tan^{2} (θ) = sec (θ)

解答

真

求解步骤

cos (θ) + cos (θ) tan^{2} (θ) = sec (θ)

调整左侧

cos (θ) + cos (θ) tan^{2} (θ)

用 sin, cos 表示

cos (θ) + cos (θ) tan^{2} (θ)

使用基本三角恒等式:

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

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

化简

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

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

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

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

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

分式相乘:

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

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

约分:

cos (θ)

= \frac{sin ^{2} ( θ )}{cos ( θ )}

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

将项转换为分式:

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

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

cos (θ) cos (θ) + sin^{2} (θ) = cos^{2} (θ) + sin^{2} (θ)

cos (θ) cos (θ) + sin^{2} (θ)

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

数字相加:

1 + 1 = 2

= cos^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

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 ( θ )}}

化简

\frac{1}{\frac{1}{s e c ( θ )}}

使用分式法则:

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

= \frac{sec ( θ )}{1}

使用法则

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

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 2cos^2(x)=1+cos(2x)

p ro v e 2 cos^{2} (x) = 1 + cos (2 x)

证明 ((tan^2(x)))/(sec(x))=sin(x)tan(x)

p ro v e \frac{( tan ^{2} ( x ) )}{sec ( x )} = sin (x) tan (x)

证明 cos(x)(1+tan(x))^2=sec(x)+2sin(x)

p ro v e cos (x) (1 + tan (x))^{2} = sec (x) + 2 sin (x)

证明 cos(x)cot(x)= 1/(sin(x))-sin(x)

p ro v e cos (x) cot (x) = \frac{1}{sin ( x )} - sin (x)

证明 tan(θ)=(sec(θ))/(csc(θ))

p ro v e tan (θ) = \frac{sec ( θ )}{csc ( θ )}