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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Factor

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

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

Factor out common term

\frac{1}{2}

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

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

Simplify

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

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

Multiply the numbers:

1 \cdot 2 = 2

= 2 cos^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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