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

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

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

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 0

sin (\frac{0}{2}) = 0

sin (\frac{0}{2})

Simplify

= sin (0)

Use the following trivial identity:

sin (0) = 0

sin (0)

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0

\frac{1 + cos ( 0 )}{2} = 1

\frac{1 + cos ( 0 )}{2}

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

= \frac{1 + 1}{2}

Simplify

= 1

The two sides are not equal

\Rightarrow False

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