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 sec^2(α)+csc^2(α)= 1/(sin^2(α)cos^2(α))

解答

证明

sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

解答

真

求解步骤

sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

调整左侧

sec^{2} (α) + csc^{2} (α)

用 sin, cos 表示

csc^{2} (α) + sec^{2} (α)

使用基本三角恒等式:

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

= (\frac{1}{sin ( α )})^{2} + sec^{2} (α)

使用基本三角恒等式:

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

= (\frac{1}{sin ( α )})^{2} + (\frac{1}{cos ( α )})^{2}

化简

(\frac{1}{sin ( α )})^{2} + (\frac{1}{cos ( α )})^{2} : \frac{cos ^{2} ( α ) + sin ^{2} ( α )}{sin ^{2} ( α ) cos ^{2} ( α )}

(\frac{1}{sin ( α )})^{2} + (\frac{1}{cos ( α )})^{2}

(\frac{1}{sin ( α )})^{2} = \frac{1}{sin ^{2} ( α )}

(\frac{1}{sin ( α )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sin ^{2} ( α )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( α )}

(\frac{1}{cos ( α )})^{2} = \frac{1}{cos ^{2} ( α )}

(\frac{1}{cos ( α )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{cos ^{2} ( α )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( α )}

= \frac{1}{sin ^{2} ( α )} + \frac{1}{cos ^{2} ( α )}

sin^{2} (α), cos^{2} (α)

的最小公倍数:

sin^{2} (α) cos^{2} (α)

sin^{2} (α), cos^{2} (α)

最小公倍数 (LCM)

计算出由出现在

sin^{2} (α)

或

cos^{2} (α)

中的因子组成的表达式

= sin^{2} (α) cos^{2} (α)

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^{2} (α) cos^{2} (α)

对于

\frac{1}{sin ^{2} ( α )} :

将分母和分子乘以

cos^{2} (α)

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

对于

\frac{1}{cos ^{2} ( α )} :

将分母和分子乘以

sin^{2} (α)

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

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

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} ( α ) + sin ^{2} ( α )}{sin ^{2} ( α ) cos ^{2} ( α )}

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{1}{cos ^{2} ( α ) sin ^{2} ( α )}

= \frac{1}{cos ^{2} ( α ) sin ^{2} ( α )}

= \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

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 sin^2(A)+cos^2(A)=1

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

证明 csc(2x)+cot(2x)=cot(x)

p ro v e csc (2 x) + cot (2 x) = cot (x)

证明 cos^2(t)=(1+cos(2t))/2

p ro v e cos^{2} (t) = \frac{1 + cos ( 2 t )}{2}

证明 cos(x)csc(x)tan(x)=1

p ro v e cos (x) csc (x) tan (x) = 1

证明 (csc^2(θ))/(1+tan^2(θ))=cot^2(θ)

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