Is cos^4(θ)= 1/4+(cos(2θ))/2+(cos^2(2θ))/4 ?

The answer to whether cos^4(θ)= 1/4+(cos(2θ))/2+(cos^2(2θ))/4 is True

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Popular Trigonometry >

prove cos^4(θ)= 1/4+(cos(2θ))/2+(cos^2(2θ))/4

Solution

prove

cos^{4} (θ) = \frac{1}{4} + \frac{cos ( 2 θ )}{2} + \frac{cos ^{2} ( 2 θ )}{4}

Solution

True

Solution steps

cos^{4} (θ) = \frac{1}{4} + \frac{cos ( 2 θ )}{2} + \frac{cos ^{2} ( 2 θ )}{4}

Manipulating right side

\frac{1}{4} + \frac{cos ( 2 θ )}{2} + \frac{cos ^{2} ( 2 θ )}{4}

Rewrite using trig identities

\frac{1}{4} + \frac{cos ( 2 θ )}{2} + \frac{cos ^{2} ( 2 θ )}{4}

Use the Double Angle identity:

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

= \frac{1}{4} + \frac{2 cos ^{2} ( θ ) - 1}{2} + \frac{( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4}

Simplify

\frac{1}{4} + \frac{2 cos ^{2} ( θ ) - 1}{2} + \frac{( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4} : cos^{4} (θ)

\frac{1}{4} + \frac{2 cos ^{2} ( θ ) - 1}{2} + \frac{( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4}

Combine the fractions

\frac{1}{4} + \frac{( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4} : \frac{1 + ( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + ( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4}

= \frac{( 2 cos ^{2} ( θ ) - 1 ) ^{2} + 1}{4} + \frac{2 cos ^{2} ( θ ) - 1}{2}

\frac{1 + ( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4} = \frac{2 cos ^{4} ( θ ) - 2 cos ^{2} ( θ ) + 1}{2}

\frac{1 + ( 2 cos ^{2} ( θ ) - 1 ) ^{2}}{4}

Expand

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

1 + (2 cos^{2} (θ) - 1)^{2}

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

Apply Perfect Square Formula:

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

a = 2 cos^{2} (θ), b = 1

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

(2 cos^{2} (θ))^{2} = 4 cos^{4} (θ)

(2 cos^{2} (θ))^{2}

Apply exponent rule:

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

= 2^{2} (cos^{2} (θ))^{2}

(cos^{2} (θ))^{2} : cos^{4} (θ)

Apply exponent rule:

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

= cos^{2 \cdot 2} (θ)

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (θ)

= 2^{2} cos^{4} (θ)

2^{2} = 4

= 4 cos^{4} (θ)

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

2 \cdot 2 cos^{2} (θ) \cdot 1

Multiply the numbers:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (θ)

= 4 cos^{4} (θ) - 4 cos^{2} (θ) + 1

= 4 cos^{4} (θ) - 4 cos^{2} (θ) + 1

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

Simplify

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

1 + 4 cos^{4} (θ) - 4 cos^{2} (θ) + 1

Group like terms

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

Add the numbers:

1 + 1 = 2

= 4 cos^{4} (θ) - 4 cos^{2} (θ) + 2

= 4 cos^{4} (θ) - 4 cos^{2} (θ) + 2

= \frac{4 cos ^{4} ( θ ) - 4 cos ^{2} ( θ ) + 2}{4}

Factor

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

4 cos^{4} (θ) - 4 cos^{2} (θ) + 2

Rewrite as

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

Factor out common term

2

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

= \frac{2 ( 2 cos ^{4} ( θ ) - 2 cos ^{2} ( θ ) + 1 )}{4}

Cancel the common factor:

2

= \frac{2 cos ^{4} ( θ ) - 2 cos ^{2} ( θ ) + 1}{2}

= \frac{2 cos ^{4} ( θ ) - 2 cos ^{2} ( θ ) + 1}{2} + \frac{2 cos ^{2} ( θ ) - 1}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ^{4} ( θ ) - 2 cos ^{2} ( θ ) + 1 + 2 cos ^{2} ( θ ) - 1}{2}

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

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

Group like terms

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

Add similar elements:

- 2 cos^{2} (θ) + 2 cos^{2} (θ) = 0

= 2 cos^{4} (θ) + 1 - 1

1 - 1 = 0

= 2 cos^{4} (θ)

= \frac{2 cos ^{4} ( θ )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos^{4} (θ)

= cos^{4} (θ)

= cos^{4} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos^4(θ)= 1/4+(cos(2θ))/2+(cos^2(2θ))/4 ?
The answer to whether cos^4(θ)= 1/4+(cos(2θ))/2+(cos^2(2θ))/4 is True