Question

Please answer,it's urgent and challenging..

Answer 1

LHS 
= (tanA + cosecB)^2 - (cotB-secA)^2 
= (tan^2A + 2tanAcosecB + cosec^2B) - (cot^2B - 2cotBsecA + sec^2A) 
= tan^2A + cosec^2B + 2tanAcosecB - cot^2B - sec^2A + 2cotBsecA 

Substituting (tan^2A = sec^2A - 1) and (cosec^2B = 1 + cot^2B) in the above step: 
(sec^2A - 1) + (1 + cot^2B) + 2tanAcosecB - cot^2B - sec^2A + 2cotBsecA 
= 2tanAcosecB + 2cotBsecA 
= 2(tanAcosecB + cotBsecA) 

RHS 
= 2tanAcotB(cosecA + secB) 
= 2tanAcotB((1 / sinA) + (1 / cosB)) 
= 2(tanA / sinA)cotB + 2tanA(cot B / cosB) 
= 2(sinA / (cosAsinB))cotB + 2tanA(cosB / (sinBcosB)) 
= 2(1 / cosA)cotB + 2 tanA(1 / sinB) 
= 2secAcotB + 2tanAcosecB 
= 2(secAcotB + tanAcosecB) 

LHS = RHS