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受欢迎的三角函数 >

证明 cos(4x)=1-8sin^2(x)cos^2(x)

解答

证明

cos (4 x) = 1 - 8 sin^{2} (x) cos^{2} (x)

解答

真

求解步骤

cos (4 x) = 1 - 8 sin^{2} (x) cos^{2} (x)

调整左侧

cos (4 x)

使用三角恒等式改写

cos (4 x)

= cos (2 \cdot 2 x)

使用倍角公式:

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

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

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

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

化简

1 - 2 (2 sin (x) cos (x))^{2} : 1 - 8 sin^{2} (x) cos^{2} (x)

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

2 (2 sin (x) cos (x))^{2} = 2^{3} sin^{2} (x) cos^{2} (x)

2 (2 sin (x) cos (x))^{2}

(2 sin (x) cos (x))^{2} = 2^{2} sin^{2} (x) cos^{2} (x)

(2 sin (x) cos (x))^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} sin^{2} (x) cos^{2} (x)

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

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2^{2} = 2^{1 + 2}

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

数字相加:

1 + 2 = 3

= 2^{3} sin^{2} (x) cos^{2} (x)

= 1 - 2^{3} sin^{2} (x) cos^{2} (x)

2^{3} = 8

= 1 - 8 sin^{2} (x) cos^{2} (x)

= 1 - 8 sin^{2} (x) cos^{2} (x)

= 1 - 8 sin^{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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