Answer
The formulas are equivalent.
Work Step by Step
Since the two LHS's are equal, we check if the RHS's are also equal.
The RHS of formula 1 =
$-\ln|\sin x|+C=\ln|\sin x|+C$
... definition of csc: $\ \ \displaystyle \csc x=\frac{1}{\sin x} =(\sin x)^{-1}$...
... so $|\sin x| =| \csc x|^{-1}$
$=\ln(|\csc x|)^{-1}+C$
...Log Property: $n\ln M=\ln M^{n}...$
$=-1\cdot\log|\csc x |+C$
$=-\log|\csc x |+C$
$... $= The RHS of formula 2
Therefore, the formulas are equivalent.