Sin2t cos2t(tg2t+ctg2t+2) can be simplified as follows:

First, we can rewrite it as:

(2sin t cos t)(2tan t cot t + 2)

Next, we can simplify the expression inside the parentheses:

2tan t cot t = 2

Therefore, our expression simplifies to:

(2sin t cos t)(2 + 2)

= 4sin t cos t

= 2sin 2t

So, Sin2t cos2t(tg2t+ctg2t+2) simplifies to 2sin 2t.