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受欢迎的三角函数 >

证明 cos^2(x)= 1/2+1/2 cos(2x)

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

分解

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

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

因式分解出通项

\frac{1}{2}

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

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

分式相乘:

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

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

1 \cdot (1 + 2 cos^{2} (x) - 1) = 2 cos^{2} (x)

1 \cdot (1 + 2 cos^{2} (x) - 1)

化简

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

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

对同类项分组

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

1 - 1 = 0

= 2 cos^{2} (x)

= 1 \cdot 2 cos^{2} (x)

数字相乘:

1 \cdot 2 = 2

= 2 cos^{2} (x)

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

数字相除:

\frac{2}{2} = 1

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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