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受欢迎的三角函数 >

证明 cos(2A)=2cos^2(A)-1

解答

证明

cos (2 A) = 2 cos^{2} (A) - 1

解答

真

求解步骤

cos (2 A) = 2 cos^{2} (A) - 1

调整左侧

cos (2 A)

使用三角恒等式改写

cos (2 A)

改写为

= cos (A + A)

使用角和恒等式:

cos (s + s) = cos (s) cos (s) - sin (s) sin (s)

= cos (A) cos (A) - sin (A) sin (A)

化简

cos (A) cos (A) - sin (A) sin (A) : cos^{2} (A) - sin^{2} (A)

cos (A) cos (A) - sin (A) sin (A)

cos (A) cos (A) = cos^{2} (A)

cos (A) cos (A)

使用指数法则:

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

cos (A) cos (A) = cos^{1 + 1} (A)

= cos^{1 + 1} (A)

数字相加:

1 + 1 = 2

= cos^{2} (A)

sin (A) sin (A) = sin^{2} (A)

sin (A) sin (A)

使用指数法则:

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

sin (A) sin (A) = sin^{1 + 1} (A)

= sin^{1 + 1} (A)

数字相加:

1 + 1 = 2

= sin^{2} (A)

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

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

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

使用毕达哥拉斯恒等式:

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

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

= cos^{2} (A) - (1 - cos^{2} (A))

化简

cos^{2} (A) - (1 - cos^{2} (A)) : 2 cos^{2} (A) - 1

cos^{2} (A) - (1 - cos^{2} (A))

- (1 - cos^{2} (A)) : - 1 + cos^{2} (A)

- (1 - cos^{2} (A))

打开括号

= - 1 - (- cos^{2} (A))

使用加减运算法则

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

= - 1 + cos^{2} (A)

= cos^{2} (A) - 1 + cos^{2} (A)

化简

cos^{2} (A) - 1 + cos^{2} (A) : 2 cos^{2} (A) - 1

cos^{2} (A) - 1 + cos^{2} (A)

对同类项分组

= cos^{2} (A) + cos^{2} (A) - 1

同类项相加:

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

= 2 cos^{2} (A) - 1

= 2 cos^{2} (A) - 1

= 2 cos^{2} (A) - 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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