The expression provided is:
sin^2$2a$ + cos^2$2a$ + ctg^2$5a$

Given that:
sin^2$\theta$ + cos^2$\theta$ = 1
ctg$\theta$ = 1/tan$\theta$

We can rewrite the expression using these identities:

sin^2$2a$ + cos^2$2a$ + ctg^2$5a$ = 1 + 1 + $1/tan(5a)$^2
= 2 + cot^2$5a$

Therefore, the simplified expression is:
2 + cot^2$5a$