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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

False

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

Show that the two sides are not equal

For

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

plug in

u = π

cos (π) = - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

= - 1

\frac{1 + cos ( 2 π )}{2} = 1

\frac{1 + cos ( 2 π )}{2}

Rewrite using trig identities:

cos (2 π) = 1

cos (2 π)

cos (2 π) = cos (0)

cos (2 π)

Rewrite

2 π

2 π + 0

= cos (2 π + 0)

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

cos (2 π + 0) = cos (0)

= cos (0)

= cos (0)

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

= 1

= \frac{1 + 1}{2}

Simplify

= 1

The two sides are not equal

\Rightarrow False

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

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

tan^{2} (A) + 1 = sec^{2} (A)

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

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

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

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