Question

(cosectheta - cottheta)^2 = 1-costheta / 1+ costheta​

Answer 1

Question

$\sf \: {(cosec \theta - cot \theta)}^{2} = \frac{1 - cos \theta}{ 1 + cos \theta}$

Formula used :

$➊ \: \: \sf \: cosec \theta = \frac{1}{sin \theta} \\ \\ ➋ \: \: \sf \: cot \theta = \frac{cos \theta}{sin \theta}$

Solution :

taking LHS, (cosecθ − cotθ)²

$\sf \: ➟ \: \: { \bigg(\frac{1}{sinθ} - \frac{cosθ}{sinθ}\bigg )}^{2} \\ \\ \bf \: taking \: LCM \\ \\ \sf \: ➟ \: \: { \bigg (\frac{1 - cosθ}{sinθ} \bigg)}^{2} \\ \\ ➟ \: \: \sf \: \frac{ {(1 -cosθ) }^{2} }{ {(sinθ)}^{2} } \\ \\ ➟ \sf \: \: using \: \:( 1 - {cosθ}^{2} = {sinθ}^{2} ) \\ \\ ➟ \: \sf \: \frac{(1 - {cosθ})^{2} }{1 - {cosθ}^{2} } \\ \\ ➟ \: \sf \: \frac{(1 - {cosθ})^{2} }{ {(1)}^{2} - {(cosθ)}^{2} } \\ \\ ➟ \: \sf \: \: \frac{ {(1 - cosθ)}^{2} }{(1 + cosθ)(1 -cosθ) } \\ \\ ➟ \: \sf \: \: \frac{1 - cosθ}{1 + cosθ}$

= RHS

LHS = RHS

Hence proved