Q.1 Prove the following identities:-

Question

(i) tan^3x/1+tan^2x + cot^3x/1+cot^2 = 1-2sin^x cos^x/sinx cosx

(ii) (1+cotx+tanx)(sinx-cosx)/sec^3x-cosec^3x = sin^2xcos^2x.

Answer 1

without brackets the first one is much too ambiguous to attempt.

even the second one, I will fix so it works

LS = (1+cotx+tanx)(sinx-cosx)/(sec^3x-cosec^3x)
= (1 + cosx/sinx + sinx/cosx)(sinx-cosx)/(sec^3x-cosec^3x)
= sinx - cosx + cosx - cos^2x + sin^2x - sinx)/(1/cos^3x - 1/sin^3x)
= ((sin^2x/cosx - cos^2x/sinx)/((sin^3x - cos^3x)/(sin^3xcos^3x))
= (sin^3x - cos^3x)/(sinxcosx) (sin^3xcos^3x)/(sin^3-cos^3x)
= sin^2xcos^2x

= RS