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受欢迎的三角函数 >

证明 1-tan^4(x)=2sec^2(x)-sec^4(x)

解答

证明

1 - tan^{4} (x) = 2 sec^{2} (x) - sec^{4} (x)

解答

真

求解步骤

1 - tan^{4} (x) = 2 sec^{2} (x) - sec^{4} (x)

调整右侧

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

分解

- sec^{4} (x) + 2 sec^{2} (x) : sec^{2} (x) (- sec^{2} (x) + 2)

- sec^{4} (x) + 2 sec^{2} (x)

使用指数法则:

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

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

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

因式分解出通项

sec^{2} (x)

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

sec^{2} (x) = tan^{2} (x) + 1

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

乘开

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

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

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

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

打开括号

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

使用加减运算法则

+ (- a) = - a

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

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

化简

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

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

对同类项分组

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

数字相加/相减:

2 - 1 = 1

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

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

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

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

乘开

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

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

使用平方差公式:

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

a = 1, b = tan^{2} (x)

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

化简

1^{2} - (tan^{2} (x))^{2} : 1 - tan^{4} (x)

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

(tan^{2} (x))^{2}

使用指数法则:

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

= tan^{2 \cdot 2} (x)

数字相乘:

2 \cdot 2 = 4

= tan^{4} (x)

= 1 - tan^{4} (x)

= 1 - tan^{4} (x)

= 1 - tan^{4} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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