Question

Prove that (tan^2A/tan^2A-1)+(cosec^2A/sec^2A-cosec^2A)=1/1-2 cos^2 A

Answer 1

Answer:

LHS = tan2Atan2A−1+cosec2Asec2A−cosec2A

= sin2Acos2Asin2Acos2A−1+1sin2A1cos2A−1sin2A

= sin2Acos2Asin2A−cos2Acos2A+1sin2Asin2A−cos2Asin2A cos2A

= sin2Asin2A−cos2A+1sin2A×sin2Acos2Asin2A−cos2A

= sin2Asin2A−cos2A+cos2Asin2A−cos2A

= sin2A+cos2Asin2A−cos2A

= 1sin2A−cos2A

= RHS

Hence proved

Answer 2

$\frac{ {tan}^{2} a}{ {tan}^{2} a - 1} + \frac{ {cosec}^{2}a }{ {sec}^{2}a - {cosec}^{2}a } \\ \\ = \frac{ {sin}^{2}a }{ {sin}^{2}a - {cos}^{2} a} + \frac{ \frac{1}{ {sin}^{2} a} }{ \frac{ {sin}^{2} a - {cos}^{2} a}{ {sin}^{2}a {cos}^{2}a } } \\ \\ = \frac{ {sin}^{2} a}{ {sin}^{2} a - {cos}^{2} a} + \frac{ {cos}^{2} a}{ {sin}^{2} a - {cos}^{2} a}$

$= \frac{1}{ {sin}^{2}a - {cos}^{2}a } \\ \\ = \frac{1}{1 - {cos}^{2}a - {cos}^{2} a} \\ \\ = \frac{1}{1 - 2 {cos}^{2} a}$