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

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

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

prove

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

True

Solution steps

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

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

Simplify

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

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

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Simplify

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

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

Group like terms

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

Add the numbers:

1 + 1 = 2

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

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

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

Factor

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

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

Rewrite as

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

Factor out common term

2

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

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

Divide the numbers:

\frac{2}{2} = 1

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

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

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

= 1 - cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan^{4} (θ) - sec^{4} (θ) = 1 - 2 sec^{2} (θ)

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

2 csc (2 x) tan (x) = sec^{2} (x)

sec (x) - tan (x) = cos (x)

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

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