How do you verify the equation is an identity?

Question

Tan^2x-tan^2y=sec^2x-sec^2y

and, how do you factor and simplify,
cscx(sin^2x+cos^2xtanx)/sinx+cosx

Answer 1

One of the basic identities is
tan^2 A + 1 = sec^2 A which gives Tan^2 A = sec^2 A - 1

so
Left side
= tan^2x-tan^2y
= sec^2x - 1 -(sec^2y - 1)
= sec^2x - sec^2y
= Right Side

Answer 2

For the second, I will assume you meant

cscx(sin^2x+cos^2xtanx)/(sinx+cosx)
= 1/sinx(sin^2x + cos^2xsinx/cosx)/(sinx + cosx)
= 1/sinx(sin^2x + sinxcosx)/(sinx+cosx)
= (sinx + cosx)/sinx + cosx)
= 1