Question

8 Tan^2 theta sin theta + 4 Tan^2 theta = 0

Answer 1

fcator out 4tan^2θ to get

4tan^2θ(2sinθ+1) = 0

so, tanθ = 0 or sinθ = -1/2

θ=0 or θ = -π/6

You didn't specify the domain, so I'll let you worry about that. If you want all solutions, then that would be

θ = πn
θ = π/6 + +2πn

for any integer n

Answer 2

fcator out 4tan^2θ to get

4tan^2θ(2sinθ+1) = 0

so, tanθ = 0 or sinθ = -1/2

θ=0 or θ = 7π/6 or θ = 11π/6

You didn't specify the domain, so I'll let you worry about that. If you want all solutions, then that would be

θ = πn
θ = 7π/6 + +2πn
θ = 11π/6 + +2πn

for any integer n