Is cos(4A)=8cos^4(A)-8cos^2(A)+1 ?

The answer to whether cos(4A)=8cos^4(A)-8cos^2(A)+1 is True

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

prove

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

True

Solution steps

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

Manipulating left side

cos (4 A)

Rewrite using trig identities

cos (4 A)

= cos (2 \cdot 2 A)

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

Apply Perfect Square Formula:

(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}

Simplify

(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}

Apply rule

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}

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

Multiply the numbers:

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

Multiply the numbers:

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

Expand

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)

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

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

Multiply the numbers:

2 \cdot 4 = 8

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

Multiply the numbers:

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

Add/Subtract the numbers:

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

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

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

Is cos(4A)=8cos^4(A)-8cos^2(A)+1 ?
The answer to whether cos(4A)=8cos^4(A)-8cos^2(A)+1 is True