How do you verify the identity cot^2t/csct=csct-sint?

Jan 17, 2017

see below

Explanation:

Left Hand Side:

${\cot}^{2} \frac{t}{\csc} t = \frac{{\cos}^{2} \frac{t}{\sin} ^ 2 t}{\frac{1}{\sin} t}$

$= {\cos}^{2} \frac{t}{\sin} ^ 2 t \cdot \sin \frac{t}{1}$

$= {\cos}^{2} \frac{t}{\sin} ^ \cancel{2} t \cdot \frac{\cancel{\sin t}}{1}$

$= {\cos}^{2} \frac{t}{\sin} t$

Use identity: ${\sin}^{2} t + {\cos}^{2} t = 1$

$= \frac{1 - {\sin}^{2} t}{\sin} t$

$= \frac{1}{\sin} t - {\sin}^{2} \frac{t}{\sin} t$

$= \frac{1}{\sin} t - {\sin}^{\cancel{2}} \frac{t}{\cancel{\sin t}}$

$= \csc t - \sin t$

$\therefore =$ Right Hand Side