Question

Sec^2A. Cosec^2= tan^2A + Cot^2A + 2 Please prove it through LHS.​

Answer 1

Answer:

#LHS=tan^2A+cot^2A+2#

#=tan^2A+1+cot^2A+1#

#=sec^2A+csc^2A#

#=1/cos^2A+1/sin^2A#

#=(sin^2A+cos^2A)/(cos^2Asin^2A)#

#=1/(cos^2Asin^2A)#

#=sec^2Acsc^2A=RHS#

Answer 2

${sec}^{2} a \times {cosec}^{2} a = {tan}^{2} a + {cot}^{2} + 2 \\ lhs = {sec}^{2} a \times {cosec}^{2} a \\ = \frac{1}{ {cos}^{2}a } \times \frac{1}{ {sin}^{2}a } \\ = \frac{1}{ {sin}^{2}a \times {cos}^{2}a } \\ \\ rhs = {tan}^{2}a + {cot}^{2}a \: + 2 \\ = \frac{ {sin}^{2} a}{ {cos}^{2} a} + \frac{ {cos}^{2} a}{ {sin}^{2}a } + 2 \\ = \frac{ {sin}^{4} a \: + {cos}^{4} a \: + 2 {sin}^{2}a. {cos}^{2} a }{ {sin}^{2} a. {cos}^{2}a } \\ = \frac{ {( {sin}^{2} a + {cos}^{2}a) }^{2} }{ {sin}^{2}a. {cos}^{2} a} \\ = \frac{1}{ {sin}^{2} a. {cos}^{2}a } \\ \\ \\ \\ \\as \: lhs \: = rhs \\ \\ \\ hence \: proved$