Question 65913
Consider the left hand side.
(sinx + cosecx)^2 + (cosx + secx)^2
= sin^2 x + cosec^2 x + 2sinx cosecx + cos^2 x + sec^2 x + 2cosx secx

= (sin^2 x + cos^2 x) + (cosec^2 x + sec^2 x) + 2 + 2 [grouping and sinx*cosecx = 1,  secx*cosx=1]

= 1 + (1 + cot^2 x + 1 + tan^2 x) + 4  [using 1 + tan^2 x = sec^2 x and so on]

= cot^2 x + cosec^2 x + 7

= Right hand side.

Hence the proof.

Good Luck!!!