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受欢迎的三角函数 >

证明 (1+cot^2(θ))tan^2(θ)=sec^2(θ)

解答

证明

(1 + cot^{2} (θ)) tan^{2} (θ) = sec^{2} (θ)

解答

真

求解步骤

(1 + cot^{2} (θ)) tan^{2} (θ) = sec^{2} (θ)

调整左侧

(1 + cot^{2} (θ)) tan^{2} (θ)

用 sin, cos 表示

(1 + cot^{2} (θ)) tan^{2} (θ)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

使用指数法则:

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

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

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

分式相乘:

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

= \frac{sin ^{2} ( θ ) ( 1 + \frac{c o s ^{2} ( θ )}{s i n ^{2} ( θ )} )}{cos ^{2} ( θ )}

化简

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

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

将项转换为分式:

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

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

乘以:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

= \frac{\frac{s i n ^{2} ( θ ) + c o s ^{2} ( θ )}{s i n ^{2} ( θ )} sin ^{2} ( θ )}{cos ^{2} ( θ )}

乘

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

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

分式相乘:

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

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

约分:

sin^{2} (θ)

= sin^{2} (θ) + cos^{2} (θ)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{1}{cos ^{2} ( θ )}

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

化简

\frac{1}{( \frac{1}{s e c ( θ )} ) ^{2}}

(\frac{1}{sec ( θ )})^{2} = \frac{1}{sec ^{2} ( θ )}

(\frac{1}{sec ( θ )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sec ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( θ )}

= \frac{1}{\frac{1}{s e c ^{2} ( θ )}}

使用分式法则:

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

= \frac{sec ^{2} ( θ )}{1}

使用法则

\frac{a}{1} = a

= sec^{2} (θ)

sec^{2} (θ)

sec^{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

证明 cot(θ)=sec(θ)csc(θ)-tan(θ)

p ro v e cot (θ) = sec (θ) csc (θ) - tan (θ)

证明 sin^2(β)csc^2(β)-sin^2(β)=cos^2(β)

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

证明 cot^2(a)-cos^2(a)=cot^2(a)cos^2(a)

p ro v e cot^{2} (a) - cos^{2} (a) = cot^{2} (a) cos^{2} (a)

证明 cos^2(θ)(sec^2(θ)-1)=sin^2(θ)

p ro v e cos^{2} (θ) (sec^{2} (θ) - 1) = sin^{2} (θ)

证明 cot^2(y)sec^2(y)=1+cot^2(y)

p ro v e cot^{2} (y) sec^{2} (y) = 1 + cot^{2} (y)