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

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

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

prove

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

True

Solution steps

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

Manipulating right side

\frac{1}{2} + \frac{1}{2} cos (4 x)

Simplify

\frac{1}{2} + cos (4 x) \frac{1}{2} : \frac{1 + cos ( 4 x )}{2}

\frac{1}{2} + cos (4 x) \frac{1}{2}

Multiply

cos (4 x) \frac{1}{2} : \frac{cos ( 4 x )}{2}

cos (4 x) \frac{1}{2}

Multiply fractions:

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

= \frac{cos ( 4 x )}{2}

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

Apply rule

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

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

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

Rewrite using trig identities

\frac{1 + cos ( 4 x )}{2}

= \frac{1 + cos ( 2 \cdot 2 x )}{2}

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (2 x)

= \frac{2 cos ^{2} ( 2 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos^{2} (2 x)

= cos^{2} (2 x)

= cos^{2} (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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