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受欢迎的三角函数 >

证明 cos^2(x)= 1/2 (cos(2x)+1)

解答

证明

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

解答

真

求解步骤

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

调整右侧

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

分式相乘:

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

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

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

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

- 1 + 1 = 0

= 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

流行的例子

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p ro v e (csc (x) - 1) (csc (x) + 1) = cot^{2} (x)

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p ro v e tan^{2} (y) (csc^{2} (y) - 1) = 1

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p ro v e cos (2 π + x) = cos (x)

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