Question

Question 5 (i)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(i) $(cosec\theta -cot\theta {)}^2=\frac{(1-cos\theta )}{(1+cos\theta )}$

Answer 1

(i) $(cosec\theta -cot\theta {)}^2=\frac{(1-cos\theta )}{(1+cos\theta )}$
$L.H.S.=(cosec\theta -cot\theta {)}^2$
$=(cose{c}^2\theta +co{t}^2\theta -2cosec\theta cot\theta )$
$=(\frac{1}{si{n}^2\theta }+\frac{co{s}^2\theta }{si{n}^2\theta }-\frac{2cos\theta }{si{n}^2\theta })$
$=\frac{(1+co{s}^2\theta -2cos\theta )}{(1-co{s}^2\theta )}$
$=\frac{(1-cos\theta {)}^2}{(1-cos\theta )(1+cos\theta )}$
$=\frac{(1-cos\theta )}{(1+cos\theta )}=R.H.S.$