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受欢迎的三角函数 >

证明 cos^2(θ/2)=(sec(θ)+1)/(2sec(θ))

解答

证明

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

解答

真

求解步骤

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

令

: u = \frac{θ}{2}

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

证明

cos^{2} (u) = \frac{sec ( 2 u ) + 1}{2 sec ( 2 u )} :

真

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

调整右侧

\frac{sec ( 2 u ) + 1}{2 sec ( 2 u )}

用 sin, cos 表示

\frac{1 + sec ( 2 u )}{2 sec ( 2 u )}

使用基本三角恒等式:

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

= \frac{1 + \frac{1}{c o s ( 2 u )}}{2 \cdot \frac{1}{c o s ( 2 u )}}

化简

\frac{1 + \frac{1}{c o s ( 2 u )}}{2 \cdot \frac{1}{c o s ( 2 u )}} : \frac{cos ( 2 u ) + 1}{2}

\frac{1 + \frac{1}{c o s ( 2 u )}}{2 \cdot \frac{1}{c o s ( 2 u )}}

乘

2 \cdot \frac{1}{cos ( 2 u )} : \frac{2}{cos ( 2 u )}

2 \cdot \frac{1}{cos ( 2 u )}

分式相乘:

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

= \frac{1 \cdot 2}{cos ( 2 u )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{cos ( 2 u )}

= \frac{1 + \frac{1}{c o s ( 2 u )}}{\frac{2}{c o s ( 2 u )}}

化简

1 + \frac{1}{cos ( 2 u )} : \frac{cos ( 2 u ) + 1}{cos ( 2 u )}

1 + \frac{1}{cos ( 2 u )}

将项转换为分式:

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

= \frac{1 \cdot cos ( 2 u )}{cos ( 2 u )} + \frac{1}{cos ( 2 u )}

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 cos ( 2 u ) + 1}{cos ( 2 u )}

乘以:

1 \cdot cos (2 u) = cos (2 u)

= \frac{cos ( 2 u ) + 1}{cos ( 2 u )}

= \frac{\frac{c o s ( 2 u ) + 1}{c o s ( 2 u )}}{\frac{2}{c o s ( 2 u )}}

分式相除:

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

= \frac{( cos ( 2 u ) + 1 ) cos ( 2 u )}{cos ( 2 u ) \cdot 2}

约分:

cos (2 u)

= \frac{cos ( 2 u ) + 1}{2}

= \frac{1 + cos ( 2 u )}{2}

= \frac{1 + cos ( 2 u )}{2}

使用三角恒等式改写

\frac{1 + cos ( 2 u )}{2}

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

= \frac{1 + 2 cos ^{2} ( u ) - 1}{2}

化简

\frac{1 + 2 cos ^{2} ( u ) - 1}{2} : cos^{2} (u)

\frac{1 + 2 cos ^{2} ( u ) - 1}{2}

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

1 + 2 cos^{2} (u) - 1

对同类项分组

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

1 - 1 = 0

= 2 cos^{2} (u)

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

数字相除:

\frac{2}{2} = 1

= cos^{2} (u)

= cos^{2} (u)

= cos^{2} (u)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

因此

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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