Can cos(theta) x sin(theta) be simplified?

Answer 1

2 sin θ cos θ = sin 2θ
sin θ cos θ = (sin 2θ) / 2

Answer 2

It appears there is not much you can do with cosA siin A where A=theta for ease of notation. When something like this occurs and you still want to rewrite the expression look for a cleaver way to multiply by 1 in the form of a fraction. In this case multiply by 1 in the form of 2/2 which does not change the value, only the form.

So, cosAsinA = (2/2)cosAsinA = [2cosAsinA]/2
= [cosAsinA + cosAsinA]/2

That last expression should ring a bell for a familiar double angel identity. In this case the identity

sin(2A) = sin(A+A) = cosAsinA + sinAcosA
= 2cosAsinA

So iin the very first expression above we can replace
2cosAsinA with sin(2A) to give

cosAsinA = 2[cosAsinA]/2 = [2cosAsinA]/2 = [sin2A]/2

Answer 3

There is a trig identity that states
sin(2*theta) = 2*sin(theta)*cos(theta)

Divide both sides by 2 and you get
(1/2)sin(2*theta) = sin(theta)*cos(theta)

The right side is what you currently have, so the answer is
(1/2)sin(2*theta)