To solve this trigonometric equation, we'll start by using the angle addition identities to simplify the expression:

2cos(2π + t) + sin(π/2 + t)
= 2(cos(2π)cos(t) - sin(2π)sin(t)) + sin(π/2)cos(t) + cos(π/2)sin(t)
= 2(cos(t) - 0sin(t)) + 1cos(t) + 0sin(t)
= 2cos(t) + cos(t)
= 3cos(t)

So the equation simplifies to:

3cos(t) = 3

Dividing both sides by 3 gives:

cos(t) = 1

Since the cosine function has a maximum value of 1, this means that t = 0 or t = 2π.

Therefore, the solutions to the equation are t = 0 and t = 2π.