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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

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

Simplify

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)

= 1 \cdot 2 cos^{2} (2 x)

Multiply the numbers:

1 \cdot 2 = 2

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