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受欢迎的三角函数 >

证明 2cos^2(a)=(1+sec(2a))cos(2a)

解答

证明

2 cos^{2} (a) = (1 + sec (2 a)) cos (2 a)

解答

真

求解步骤

2 cos^{2} (a) = (1 + sec (2 a)) cos (2 a)

调整右侧

(1 + sec (2 a)) cos (2 a)

用 sin, cos 表示

(1 + sec (2 a)) cos (2 a)

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

将项转换为分式:

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

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

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

乘以:

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

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

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

分式相乘:

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

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

约分:

cos (2 a)

= cos (2 a) + 1

= cos (2 a) + 1

= 1 + cos (2 a)

使用三角恒等式改写

1 + cos (2 a)

使用倍角公式:

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

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

化简

1 + 2 cos^{2} (a) - 1 : 2 cos^{2} (a)

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

对同类项分组

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

1 - 1 = 0

= 2 cos^{2} (a)

= 2 cos^{2} (a)

= 2 cos^{2} (a)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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