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受欢迎的三角函数 >

证明 1+cos^4(x)-sin^4(x)=2cos^2(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

1 + cos^{4} (x) - sin^{4} (x)

分解

1 - sin^{4} (x) : - (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

1 - sin^{4} (x)

因式分解出通项

- 1

= - (sin^{4} (x) - 1)

分解

sin^{4} (x) - 1 : (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

sin^{4} (x) - 1

将

sin^{4} (x) - 1

改写为

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

sin^{4} (x) - 1

将

1

改写为

1^{2}

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

使用指数法则:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

分解

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

sin^{2} (x) - 1

将

1

改写为

1^{2}

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (x) + 1) (sin (x) - 1)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

乘开

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

(sin (x) + 1) (sin (x) - 1)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = sin (x), b = 1

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

使用法则

1^{a} = 1

1^{2} = 1

= sin^{2} (x) - 1

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

使用毕达哥拉斯恒等式:

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

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

= cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x))

化简

cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x)) : 2 cos^{2} (x)

cos^{4} (x) - (- cos^{2} (x)) (2 - cos^{2} (x))

使用法则

- (- a) = a

= cos^{4} (x) + cos^{2} (x) (2 - cos^{2} (x))

乘开

cos^{2} (x) (2 - cos^{2} (x)) : 2 cos^{2} (x) - cos^{4} (x)

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

使用分配律:

a (b - c) = ab - a c

a = cos^{2} (x), b = 2, c = cos^{2} (x)

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

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

cos^{2} (x) cos^{2} (x) = cos^{4} (x)

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

使用指数法则:

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

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

= cos^{2 + 2} (x)

数字相加:

2 + 2 = 4

= cos^{4} (x)

= 2 cos^{2} (x) - cos^{4} (x)

= cos^{4} (x) + 2 cos^{2} (x) - cos^{4} (x)

同类项相加:

cos^{4} (x) - cos^{4} (x) = 0

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 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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