## A Trigonometric Sequence

A particular sequence takes the form
$\frac{1}{sin \theta} - sin \theta , \: cos \theta , \: sin \theta , \: \frac{1}{cos \theta}- cos \theta ,...$

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
$r=\frac{sin \theta}{cos \theta } = tan \theta \lt 1 \rightarrow \theta \lt \frac{\pi}{4}$
.