demonstrate to m cosec^4(x) -cotan^4(x)=cosec^2(x) +co tan^2(x)

first step is to determine which side to simplify, first obvious thing to do is to relate to this: (a^4 - b^4) = (a^2 - b^2) * (a^2 + b^2),
so we have to simplify the left side.

cosec^4(x) -cotan^4(x) = (cosec^2(x) -cotan^2(x)) * (cosec^2(x) +cotan^2(x))
now back to the equality required to prove, we have to work with (cosec^2(x) -cotan^2(x)) and show its equal to 1!

(cosec^2(x) -cotan^2(x)) = (1/sin^2(x)) - (cos^2(x)/sin^2(x))
= (1 - cos^2(x))/sin^2(x)
= sin^2(x)/sin^2(x)
= 1

thus, cosec^4(x) -cotan^4(x) = (cosec^2(x) -cotan^2(x)) * (cosec^2(x) +cotan^2(x)) = 1 * (cosec^2(x) +cotan^2(x))

and cosec^4(x) -cotan^4(x)=cosec^2(x) +co tan^2(x). (proved)