The correct option is C If sinθ∈Q and cosθ∈Q, then tan3θ∈Q (if defined)
tanθ=1−cos2θsin2θ∈Q
As cos2θ∈Q and sin2θ∈Q,
∴tanθ∈Q
sin2θ=2tanθ1+tan2θ
cos2θ=1−tan2θ1+tan2θ
tan2θ=2tanθ1−tan2θ
If tanθ∈Q, then we can clearly conclude from the above formulae that sin2θ,cos2θ and tan2θ∈Q
If sinθ,cosθ∈Q,
then sin3θ=3sinθ−4sin3θ∈Q
and cos3θ=4cos3θ−3cosθ∈Q
∴tan3θ=sin3θcos3θ∈Q
If sinθ∈Q,
then sin3θ=3sinθ−4sin3θ∈Q
But cos3θ=±√1−sin23θ need not to be rational.
For e.g. sinθ=13∈Q
sin3θ=3×13−4×(13)3=2327
But, cos3θ=±√20027∉Q