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受欢迎的三角函数 >

证明 5sec^2(x)+4=9sec^2(x)-4tan^2(x)

解答

证明

5 sec^{2} (x) + 4 = 9 sec^{2} (x) - 4 tan^{2} (x)

解答

真

求解步骤

5 sec^{2} (x) + 4 = 9 sec^{2} (x) - 4 tan^{2} (x)

调整右侧

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= 9 sec^{2} (x) - 4 (sec^{2} (x) - 1)

化简

9 sec^{2} (x) - 4 (sec^{2} (x) - 1) : 5 sec^{2} (x) + 4

9 sec^{2} (x) - 4 (sec^{2} (x) - 1)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - 4, b = sec^{2} (x), c = 1

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

使用加减运算法则

- (- a) = a

= - 4 sec^{2} (x) + 4 \cdot 1

数字相乘:

4 \cdot 1 = 4

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

= 9 sec^{2} (x) - 4 sec^{2} (x) + 4

同类项相加:

9 sec^{2} (x) - 4 sec^{2} (x) = 5 sec^{2} (x)

= 5 sec^{2} (x) + 4

= 5 sec^{2} (x) + 4

= 5 sec^{2} (x) + 4

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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