Answer
The identity is verified.
$\frac{csc~θ}{2~cos~θ}=csc~2θ$
Work Step by Step
Use: $2~sin~θ~cos~θ=sin~2θ$ and $csc~θ=\frac{1}{sin~θ}$
Start on the right side:
$\frac{csc~θ}{2~cos~θ}=csc~θ~\frac{1}{2~cos~θ}=\frac{1}{sin~θ}~\frac{1}{2~cos~θ}=\frac{1}{2~sin~θ~cos~θ}=\frac{1}{sin~2θ}=csc~2θ$
