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受欢迎的三角函数 >

证明 sin^4(x)-cos^4(x)=2sin^2(x)-1

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

分解

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

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

将

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

改写为

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

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

使用指数法则:

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

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

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

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

分解

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

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

使用平方差公式:

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

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

= (sin (x) + cos (x)) (sin (x) - cos (x))

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

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

使用三角恒等式改写

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

乘开

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

(sin (x) + cos (x)) (sin (x) - cos (x))

使用平方差公式:

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

a = sin (x), b = cos (x)

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

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

使用倍角公式:

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

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

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

化简

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

使用倍角公式:

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

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

使用毕达哥拉斯恒等式:

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

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

化简

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

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

乘以:

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

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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