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

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

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

prove

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

True

Solution steps

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

Let:

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

We showed that the two sides could take the same form

\Rightarrow True

sin (β) = \frac{1}{2}

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

3 cos (t) - 3 cos (3 t) = 12 sin^{2} (t) cos (t)

sin (x) (tan^{2} (x) + 1) = tan (x) sec (x)

csc (u) = 2

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