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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\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

\frac{1}{2 cos ( x ) sin ( x )} = csc (2 x)

cos (x + \frac{π}{2}) = sin (x)

\frac{sec ( x ) - cos ( x )}{cos ( x )} = tan^{2} (x)

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

tan (\frac{π}{2} - x) sin (x) = cos (x)

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