Is sin(x)=2sin(x/2)cos(x/2) ?

The answer to whether sin(x)=2sin(x/2)cos(x/2) is True

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prove sin(x)=2sin(x/2)cos(x/2)

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

sin (2 u) = 2 sin (u) cos (u)

Prove

sin (2 u) = 2 sin (u) cos (u) :

True

sin (2 u) = 2 sin (u) cos (u)

Manipulating left side

sin (2 u)

Rewrite using trig identities

sin (2 u)

Rewrite as

= sin (u + u)

Use the Angle Sum identity:

sin (s + s) = sin (s) cos (s) + cos (s) sin (s)

= sin (u) cos (u) + cos (u) sin (u)

= sin (u) cos (u) + cos (u) sin (u)

Simplify

sin (u) cos (u) + cos (u) sin (u) : 2 sin (u) cos (u)

sin (u) cos (u) + cos (u) sin (u)

Add similar elements:

sin (u) cos (u) + cos (u) sin (u) = 2 sin (u) cos (u)

= 2 sin (u) cos (u)

= 2 sin (u) cos (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

Is sin(x)=2sin(x/2)cos(x/2) ?
The answer to whether sin(x)=2sin(x/2)cos(x/2) is True