Let $F<E$ and $F<K$ be finite extensions and assume that $EK$ is defined as composite of two fields. I need to show that $[EK:F] \leq [E:F][K:F]$, with equality if $[E:F]$ and $[K:F]$ are relatively prime.
I am stuck with even the less than or equal to part, and also I couldn't see the thing about the case being these two relatively prime.
Divide the inequality by one of the factors on the right; you should be able to subsequently simplify the left-hand side by using the product formula for towers. Use the PET for the numerator field on the right-hand side, so that both sides of the inequality turn into the degree of an algebraic element only the one on the left side is considered over a larger base field...
For the second part, remember your knowledge of arithmetic and elementary number theory: in particular recall that $a,b\mid n$ and $\gcd(a,b)=1$ imply $ab\mid n$. What are $a,b,n$ in this problem?