\[\frac{1}{sin \theta} - sin \theta , \: cos \theta , \: sin \theta , \: \frac{1}{cos \theta}- cos \theta ,...\]
We can simplify each term, obtaining the sequence
\[\frac{1-sin^2 \theta}{sin \theta} =\frac{cos^2 \theta}{sin \theta} , \: cos \theta , \: sin \theta , \: \frac{1-cos^2 \theta}{cos \theta}=\frac{sin^2 \theta}{cos \theta} ,...\]
This is a geometric sequence with first term
\[a=\frac{cos^2 \theta}{sin \theta}\]
and common ratio \[r=\frac{sin \theta}{cos \theta}\]
.The
\[n^{th}\]
term is \[a_N=ar^{n-1}=\frac{cos^2 \theta}{sin \theta} (\frac{sin \theta}{cos \theta})^n\]
and sum \[S=\frac{a}{1-r}=\frac{cos^2 \theta / sin \theta }{1- sin \theta / cos \theta}=\frac{cos^3 \theta}{sin \theta cos \theta - sin^2 \theta}\]
.The expression for the sum of the series is only valid for
\[r=\frac{sin \theta}{cos \theta } = tan \theta \lt 1 \rightarrow \theta \lt \frac{\pi}{4}\]
.