Question

2 root 2 sin 2t + 2 root 3 tan 2t = 0

Answer 1

2√2 sin(2t) + 2√3tan(2t) = 0
let 2t = x
2√2sinx + 2√3sinx/cosx = 0
(2√2 sinxcosx + 2√3sinx)/cosx = 0
times cosx
2√2sinxcosx + 2√3sinx = 0
2sinx(√2cosx + √3) = 0
sinx = 0 or cosx = -√3/√2

You don't say if your angle is in degrees or radians
I will use degrees, there should be no problem for you repeat it using radians.

if sinx = 0 ---> sin(2t) = 0
2t = 0 or 2t = 180 or 2t = 360
t = 0 , 90 , 180 , ....
if sinx = √3/√2, sin(2t) = -√3/√2 < - 1
so that part has no solution