1. ## Trig Identities

In a question I was given, in order to solve it, I needed to simplify $\displaystyle \sin\left(\frac{3}{2}x\right)\cos\left(\frac{3}{2} x\right)$ to $\displaystyle \frac{1}{2}\sin 3x$. This is "obvious" related to the identity $\displaystyle 2\sin x \cos x = \sin 2x$.?

So my question is, what's the rule (if any) to simplify these forms of expressions?

Lets use the example of $\displaystyle \sin\left(\frac{3}{\sqrt{2}}x\right)\cos\left(\fra c{3}{\sqrt{2}}x\right)$ or $\displaystyle \sin\left(\frac{4}{5}x\right)\cos\left(\frac{4}{5} x\right)$, what do those simplify to?

2. Hello,

This is "obvious" related to the identity
Yes

Originally Posted by polymerase
$\displaystyle \sin\left(\frac{3}{\sqrt{2}}x\right)\cos\left(\fra c{3}{\sqrt{2}}x\right)$
It depends on what purpose, sometimes, it's easier not to simplify.

$\displaystyle =\frac{1}{2} \sin \left(2 \cdot \frac{3}{\sqrt{2}} x \right)=\frac{1}{2} \sin \left(3 \sqrt{2} \ x \right)$

Is it what you want ?

or $\displaystyle \sin\left(\frac{4}{5}x\right)\cos\left(\frac{4}{5} x\right)$, what do those simplify to?
$\displaystyle =\frac 12 \sin \left(\frac{8}{5} x \right) ?$