To solve this trigonometric equation, we can use the sum-to-product identities:

sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2)

Using the first identity, we rewrite sin(pi+t) as sin(pi/2 + t) + sin(pi/2 - t), and cos(pi/2+t) as cos(pi/2 + t) + cos(pi/2 - t). Substituting these expressions into the equation:

2sin((pi+2t)/2)cos((pi-2t)/2) + 2cos((pi+2t)/2)cos((pi-2t)/2) = 3

Simplifying further:

2sin((pi+2t)/2)cos((pi-2t)/2) + 2cos((pi+2t)/2)cos((pi-2t)/2) = 3
sin(pi/2+t) = 3/2
1 = 3/2 [since sin(pi/2+t) = 1]
There is no solution to the equation sin(pi+t) + 2cos (pi/2+t) = 3.