Question

prove that (cot-cosec)2= 1-cos/1+cos​

Answer 1

${ \sf{ \large {\underline {\underline{\purple{given : }}}}}}\\ \\ { \sf {\underline{\purple{prove \: that : }}}} \\ \\ \sf \purple{( \cot \: \theta - \cosec \theta) {}^{2} = ( \frac{1 - cos}{1 + cos \: \theta} )} \\ \\ \\ { \sf{ \large {\underline {\underline{\pink{solution : }}}}}}\\ \\ \tt \: = \: ( \cot \: \theta \: - \: cosec \: \theta \: ) {}^{2} \\ \\ = \tt \: ( \frac{ \cos \: \theta }{ sin} \: - \: \frac{1}{sin} ) {}^{2} \: \: = \: \: ( \frac{cos \: \theta \: - \: 1}{sin \: \theta} ) {}^{2} \\ \\ \tt \: = \frac{(1 - cos \: \theta) {}^{2} }{(sin {}^{2} \: \theta) } \: \: \: ( \therefore \: sin {}^{2} \theta \: + \: cos {}^{2} \: \theta \: = \: 1) \\ \\ \tt \: = \: \frac{(1 - cos \: \theta) {}^{2} }{(1 - cos {}^{2} \: \theta)} \\ \\ \tt \: = \: \frac{(1 - \cos \theta)(1 - cos \: \theta) }{(1 - cos \: \theta)(1 + cos \: \theta)} \: \\ \\ \tt\: = \: \frac{1 - cos \: \theta}{1 + cos \: \theta} \\ \\ \tt \pink{ LHS = RHS} \\ \\ \tt \pink{hence \: \: proved.}$