Is 4sin(x/4)cos(x/4)cos(x/2)=sin(x) ?

The answer to whether 4sin(x/4)cos(x/4)cos(x/2)=sin(x) is True

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prove 4sin(x/4)cos(x/4)cos(x/2)=sin(x)

prove

4 sin (\frac{x}{4}) cos (\frac{x}{4}) cos (\frac{x}{2}) = sin (x)

True

Solution steps

4 sin (\frac{x}{4}) cos (\frac{x}{4}) cos (\frac{x}{2}) = sin (x)

Let:

u = \frac{x}{4}

4 sin (u) cos (u) cos (2 u) = sin (4 u)

Prove

4 sin (u) cos (u) cos (2 u) = sin (4 u) :

True

4 sin (u) cos (u) cos (2 u) = sin (4 u)

Manipulating left side

4 sin (u) cos (u) cos (2 u)

Rewrite using trig identities

4 sin (u) cos (u) cos (2 u)

= 2 \cdot 2 sin (u) cos (u) cos (2 u)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= sin (2 \cdot 2 u)

Simplify

= sin (4 u)

= sin (4 u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

4 sin (\frac{x}{4}) cos (\frac{x}{4}) cos (\frac{x}{2}) = sin (x)

We showed that the two sides could take the same form

\Rightarrow True

Is 4sin(x/4)cos(x/4)cos(x/2)=sin(x) ?
The answer to whether 4sin(x/4)cos(x/4)cos(x/2)=sin(x) is True