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受欢迎的三角函数 >

证明 cos(4A)=8cos^4(A)-8cos^2(A)+1

解答

证明

cos (4 A) = 8 cos^{4} (A) - 8 cos^{2} (A) + 1

解答

真

求解步骤

cos (4 A) = 8 cos^{4} (A) - 8 cos^{2} (A) + 1

调整左侧

cos (4 A)

使用三角恒等式改写

cos (4 A)

= cos (2 \cdot 2 A)

使用倍角公式:

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

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

使用倍角公式:

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

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

化简

2 (2 cos^{2} (A) - 1)^{2} - 1 : 8 cos^{4} (A) - 8 cos^{2} (A) + 1

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

(2 cos^{2} (A) - 1)^{2} : 4 cos^{4} (A) - 4 cos^{2} (A) + 1

使用完全平方公式:

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

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

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

化简

(2 cos^{2} (A))^{2} - 2 \cdot 2 cos^{2} (A) \cdot 1 + 1^{2} : 4 cos^{4} (A) - 4 cos^{2} (A) + 1

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

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

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

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

(cos^{2} (A))^{2} : cos^{4} (A)

使用指数法则:

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

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

数字相乘:

2 \cdot 2 = 4

= cos^{4} (A)

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

2^{2} = 4

= 4 cos^{4} (A)

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

2 \cdot 2 cos^{2} (A) \cdot 1

数字相乘:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (A)

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

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

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

乘开

2 (4 cos^{4} (A) - 4 cos^{2} (A) + 1) : 8 cos^{4} (A) - 8 cos^{2} (A) + 2

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

打开括号

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

使用加减运算法则

+ (- a) = - a

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

化简

2 \cdot 4 cos^{4} (A) - 2 \cdot 4 cos^{2} (A) + 2 \cdot 1 : 8 cos^{4} (A) - 8 cos^{2} (A) + 2

2 \cdot 4 cos^{4} (A) - 2 \cdot 4 cos^{2} (A) + 2 \cdot 1

数字相乘:

2 \cdot 4 = 8

= 8 cos^{4} (A) - 8 cos^{2} (A) + 2 \cdot 1

数字相乘:

2 \cdot 1 = 2

= 8 cos^{4} (A) - 8 cos^{2} (A) + 2

= 8 cos^{4} (A) - 8 cos^{2} (A) + 2

= 8 cos^{4} (A) - 8 cos^{2} (A) + 2 - 1

数字相加/相减:

2 - 1 = 1

= 8 cos^{4} (A) - 8 cos^{2} (A) + 1

= 8 cos^{4} (A) - 8 cos^{2} (A) + 1

= 8 cos^{4} (A) - 8 cos^{2} (A) + 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 cos(4B)=1-8sin^2(B)+8sin^4(B)

p ro v e cos (4 B) = 1 - 8 sin^{2} (B) + 8 sin^{4} (B)

证明 sin(pi/2-u)=cos(u)

p ro v e sin (\frac{π}{2} - u) = cos (u)

证明 arccos(x)= 1/(cos(x))

p ro v e arccos (x) = \frac{1}{cos ( x )}

证明 (csc^2(x)-1)/(cos(x))=cot(x)csc(x)

p ro v e \frac{csc ^{2} ( x ) - 1}{cos ( x )} = cot (x) csc (x)

证明 cos^2(x)(1-sec^2(x))=-sin^2(x)

p ro v e cos^{2} (x) (1 - sec^{2} (x)) = - sin^{2} (x)