Verified answer

sin(9t)sin(6t) + cos(9t)cos(6t) = 0.

But we have the trig identity sin(a)sin(b) + cos(a)cos(b) = cos(a-b).

If we use that trig identity, we have cos(9t-6t) = 0, or cos(3t) = 0.

And remember cos(π/2) = 0, as does cos(3π/2) as does cos(5π/2).

So either 3t = π/2 (and therefore t = π/6), or 3t = 3π/2 (and therefore t = π/2) or 3t = 5π/2 (and therefore t = 5π/6.

So your 3 solutions are π/6, π/2, 5π/6.