Is (1-cos(2x))/2 =sin^2(x) ?

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

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prove (1-cos(2x))/2 =sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

\frac{1 - ( 1 - 2 sin ^{2} ( x ) )}{2}

Expand

1 - (1 - 2 sin^{2} (x)) : 2 sin^{2} (x)

1 - (1 - 2 sin^{2} (x))

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

- (1 - 2 sin^{2} (x))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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