if cosec theta - sin theta = m and sec theta - cos theta = n find the relation between m and n by eliminating theta

m = cosecA - sinA

= 1/sinA - sinA 

= 1-sin2A / sinA

= cos2A / sinA

m2 = cos4A / sin2A

n = secA - cosA 

= 1/cosA - cosA

= 1-cos2A / cosA

sin2A / cosA

n2 = sin4A / cos2A 

m2n2= (cos4A / sin2A) * (sin4A / cos2A)

= cos2A sin2A

m2+n2+3 = (cos4A/sin2A) + (sin4A/cos2A) + 3

= sin6A + cos6A / sin2A cos2A + 3

= (sin2A)3 + (cos2A)3 / sin2Acos2A + 3

= (sin2A + cos2A) ( sin4A + cos4A -  sin2Acos2A ) /  sin2Acos2A + 3

= (sin2A)2 + (cos2A)2 -  sin2Acos2A /  sin2Acos2A + 3

= (sin2A + cos2A)2 - 2 sin2Acos2A -  sin2Acos2A /  sin2Acos2A + 3

= 1 - 3 sin2Acos2A + 3 sin2Acos2A / sin2Acos2A

= 1 / sin2Acos2A

m2n2(m2+n2+3) = ( sin2Acos2A ) * ( 1/  sin2Acos2A ) 

= 1