Question

Can someone please help me with proving this trig identity:

sin(x/2)cos(x/2)=1/2sin(x)

Answer 1

\(\large\color{black}{ \displaystyle \sin(x/2)\cos(x/2)=\frac{1}{2} \sin(x)}\)

multiply both sides times 2, and you get:

\(\large\color{black}{ \displaystyle 2\sin(x/2)\cos(x/2)=\sin(x)}\)

and then you know the following rule (in red):

\(\large\color{red}{ \displaystyle \sin(2a)=2\sin(a)\cos(a)}\)

Answer 2

except that in your case, "a" is "x/2"

Answer 3

oh thanks I wasn't sure if I was allowed to multiply both sides when proving, but this helped.

Answer 4

Well, in general "prove" means that you can play with both sides at the same time, and "verify" means that you have to show that sides are equivalent by manipulating only on one side.

Or that is at least how I have been taught to distinguish between the two terms.

Answer 5

You don't need to do that though, if you want to as I used the term, "verify", then this is what you can do:

\(\large\color{black}{ \displaystyle \sin(x/2)\cos(x/2)=\frac{1}{2} \sin(x)}\)

\(\large\color{black}{ \displaystyle \frac{2\sin(x/2)\cos(x/2)}{2}=\frac{1}{2} \sin(x)}\)

\(\large\color{black}{ \displaystyle \frac{\sin(x/2+x/2)}{2}=\frac{1}{2} \sin(x)}\)

\(\large\color{black}{ \displaystyle \frac{1}{2} \sin(x)=\frac{1}{2} \sin(x)}\)