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受欢迎的三角函数 >

证明 4cos^2(a)+4sin^2(a)=4

解答

证明

4 cos^{2} (a) + 4 sin^{2} (a) = 4

解答

真

求解步骤

4 cos^{2} (a) + 4 sin^{2} (a) = 4

调整左侧

4 cos^{2} (a) + 4 sin^{2} (a)

使用三角恒等式改写

4 cos^{2} (a) + 4 sin^{2} (a)

使用毕达哥拉斯恒等式:

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

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

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

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

4 cos^{2} (a) + 4 (1 - cos^{2} (a))

乘开

4 (1 - cos^{2} (a)) : 4 - 4 cos^{2} (a)

4 (1 - cos^{2} (a))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (a)

= 4 \cdot 1 - 4 cos^{2} (a)

数字相乘:

4 \cdot 1 = 4

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

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

化简

4 cos^{2} (a) + 4 - 4 cos^{2} (a) : 4

4 cos^{2} (a) + 4 - 4 cos^{2} (a)

对同类项分组

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

同类项相加:

4 cos^{2} (a) - 4 cos^{2} (a) = 0

= 4

= 4

= 4

= 4

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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