Answer
We want to verify the following identity:
$ln| tan\ x sin\ x|=2 ln|sin\ x| + ln|sec\ x|$
Work Step by Step
$ln| tan\ x sin\ x|=2 ln|sin\ x| + ln|sec\ x|$
$ln| tan\ x sin\ x|=2 ln|sin\ x| + ln|sec\ x|$
$ln| tan\ x sin\ x|= ln|sin^2\ x| + ln|sec\ x|$, since $aln\ b=ln\ b^a$
$ln| tan\ x sin\ x|= ln|sin^2\ x\ sec\ x |$, since $ln\ a+ln\ b=ln\ ab$
$ln| tan\ x sin\ x|= ln|\frac{sin^2\ x}{cos\ x}|$, since $sec\ x=\frac{1}{cos\ x}$
$ln| tan\ x sin\ x|= ln|\frac{sin\ x}{cos\ x}sin\ x|$
$ln| tan\ x sin\ x|= ln|tan\ xsin\ x|$, since $tan\ x=\frac{sin\ x}{cos\ x}$
The right-hand side and left-hand side are equal, thus verifying the identity.
