I have been teaching myself calculus and in the process came across this equation:- 1/2*sin x/2*cos x/2. Can somebody show me how to simplify it? I am sure it involves the identity sin 2x=2*sin x*cos x.
Are you asking to simplify
$\displaystyle \frac{1}{2}\sin{\frac{x}{2}}\cos{\frac{x}{2}}$?
Yes, you do use the identity $\displaystyle \sin{2\theta} = 2\sin{\theta}\cos{\theta}$.
If you let $\displaystyle \theta = \frac{x}{2}$
we have $\displaystyle \frac{1}{2}\sin{\theta}\cos{\theta}$
$\displaystyle = \frac{1}{4}\cdot 2\sin{\theta}\cos{\theta}$
$\displaystyle = \frac{1}{4}\sin{2\theta}$
$\displaystyle = \frac{1}{4}\sin{\frac{2x}{2}}$
$\displaystyle = \frac{1}{4}\sin{x}$.
I won't tell you how long I stuffed around with this. Tricky little bugger - but not so difficult when somebody shows you how. Text books that just give you the answer without showing you how it was arrived at drive me nuts.
The solutions are given but sometimes the workings are very brief (important chunks missed out). In most cases I have been able to figure it out with the help of google. However it would be a time saver if everything was in the answer given without having to go searching on the internet.