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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (u)

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

Divide the numbers:

\frac{2}{2} = 1

= cos^{2} (u)

= cos^{2} (u)

= cos^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( x ) + cos ( x )}{sin ( x )} = 1 + cot (x)

sin^{2} (θ) + 2 cos^{2} (θ) - 1 = cos^{2} (θ)

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

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

cos (θ) = - cos (- θ)

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