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受欢迎的三角函数 >

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

解答

证明

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

解答

真

求解步骤

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

调整左侧

\frac{sec ( α )}{1 - cos ( α )}

使用三角恒等式改写

\frac{sec ( α )}{1 - cos ( α )}

乘以

\frac{1 + cos ( α )}{1 + cos ( α )}

= \frac{sec ( α )}{\frac{( 1 - c o s ( α ) ) ( 1 + c o s ( α ) )}{1 + c o s ( α )}}

使用平方差公式:

α^{2} - y^{2} = (α + y) (α - y)

(1 - cos (α)) (1 + cos (α)) = 1 - cos^{2} (α)

= \frac{sec ( α )}{\frac{1 - c o s ^{2} ( α )}{1 + c o s ( α )}}

使用毕达哥拉斯恒等式:

1 = cos^{2} (α) + sin^{2} (α)

1 - cos^{2} (α) = sin^{2} (α)

= \frac{sec ( α )}{\frac{s i n ^{2} ( α )}{1 + c o s ( α )}}

使用分式法则:

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

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

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

乘开

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

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

乘开

(1 + cos (α)) sec (α) : sec (α) + sec (α) cos (α)

(1 + cos (α)) sec (α)

= sec (α) (1 + cos (α))

使用分配律:

a (b + c) = ab + a c

a = sec (α), b = 1, c = cos (α)

= sec (α) \cdot 1 + sec (α) cos (α)

= 1 \cdot sec (α) + sec (α) cos (α)

乘以:

1 \cdot sec (α) = sec (α)

= sec (α) + sec (α) cos (α)

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

使用分式法则:

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

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

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

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

使用三角恒等式改写

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

sec (α) cos (α) = 1

sec (α) cos (α)

用 sin, cos 表示

sec (α) cos (α)

使用基本三角恒等式:

sec (α) = \frac{1}{cos ( α )}

= \frac{1}{cos ( α )} cos (α)

\frac{1}{cos ( α )} cos (α) = 1

\frac{1}{cos ( α )} cos (α)

分式相乘:

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

= \frac{1 cos ( α )}{cos ( α )}

约分:

cos (α)

= 1

= 1

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

使用法则

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

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

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

= \frac{sec ( α ) + 1}{sin ^{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 cos(pi/4)=(sqrt(2))/2

p ro v e cos (\frac{π}{4}) = \frac{2}{2}

证明 cot(x)=(csc(x))/(sec(x))

p ro v e cot (x) = \frac{csc ( x )}{sec ( x )}

证明 1/(1+sin(x))+1/(1-sin(x))=2+2tan^2(x)

p ro v e \frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} = 2 + 2 tan^{2} (x)

证明 sec(2x)=(sec^2(x))/(1-tan^2(x))

p ro v e sec (2 x) = \frac{sec ^{2} ( x )}{1 - tan ^{2} ( x )}

证明 (sin(3x)-sin(5x))/(2cos(4x))=-sin(x)

p ro v e \frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)