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

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

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Popular Trigonometry >

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

cos (4 x)

Rewrite using trig identities

cos (4 x)

= cos (2 \cdot 2 x)

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

2 (1 - 2 sin^{2} (x))^{2} - 1 : 8 sin^{4} (x) - 8 sin^{2} (x) + 1

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

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

Apply Perfect Square Formula:

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

a = 1, b = 2 sin^{2} (x)

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

Simplify

1^{2} - 2 \cdot 1 \cdot 2 sin^{2} (x) + (2 sin^{2} (x))^{2} : 1 - 4 sin^{2} (x) + 4 sin^{4} (x)

1^{2} - 2 \cdot 1 \cdot 2 sin^{2} (x) + (2 sin^{2} (x))^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

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

2 \cdot 1 \cdot 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 \cdot 2 = 4

= 4 sin^{2} (x)

(2 sin^{2} (x))^{2} = 4 sin^{4} (x)

(2 sin^{2} (x))^{2}

Apply exponent rule:

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

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

(sin^{2} (x))^{2} : sin^{4} (x)

Apply exponent rule:

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

= sin^{2 \cdot 2} (x)

Multiply the numbers:

2 \cdot 2 = 4

= sin^{4} (x)

= 2^{2} sin^{4} (x)

2^{2} = 4

= 4 sin^{4} (x)

= 1 - 4 sin^{2} (x) + 4 sin^{4} (x)

= 1 - 4 sin^{2} (x) + 4 sin^{4} (x)

= 2 (1 - 4 sin^{2} (x) + 4 sin^{4} (x)) - 1

Expand

2 (1 - 4 sin^{2} (x) + 4 sin^{4} (x)) : 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

2 (1 - 4 sin^{2} (x) + 4 sin^{4} (x))

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

2 \cdot 1 - 2 \cdot 4 sin^{2} (x) + 2 \cdot 4 sin^{4} (x) : 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

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

Multiply the numbers:

2 \cdot 1 = 2

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

Multiply the numbers:

2 \cdot 4 = 8

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x)

= 2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1

Simplify

2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1 : 8 sin^{4} (x) - 8 sin^{2} (x) + 1

2 - 8 sin^{2} (x) + 8 sin^{4} (x) - 1

Group like terms

= - 8 sin^{2} (x) + 8 sin^{4} (x) + 2 - 1

Add/Subtract the numbers:

2 - 1 = 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 8 sin^{4} (x) - 8 sin^{2} (x) + 1

= 1 - 8 sin^{2} (x) + 8 sin^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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