Question

Cosec A+1/ cosec A- 1 = (sec A+ tan A)²

Answer 1

proof:
LHS= cosec A + 1/ cosec A - 1
=(cosecA+1/cosecA-1)x(cosec A+1/ cosecA+1) (by taking conjugate)
=>[(cosec A + 1)^ 2/ (cosec^ 2 A - 1)]
=>[(cosec^2+1+2cosecA)/cot^2A] (cosec^2A-1=cot^2A)
=>[(cot^2A+2+2cosecA)/(1/tan^2A)
=>[(1/tan^2A+2(1+cosecA))/tan^2A]x(tan^2A)/1
=>[{1+2tan^2A(1+cosecA)}]
=>[1+2tan^2A+2tan^2AcosecA]
=>(1+2sin^2A/cos^2A+2sin^2AcosecA/cos^2A)
=>(1+{2sin^2A}{1/cos^2A}+{2sin^2A(1/ sinA)}{1/cos^2A})
=>(1+{2sin^2A}{sec^2A}+{2sinA}{sec^2A})
=>(1+2sin^2Asec^2A+2sinAsec^2A)
=>(secA+tanA)^2=RHS