Answer:
Option C
Explanation:
sin9θcos27θ+sin3θcos9θ+sinθcos3θ=k(tan27θ−tanθ)
LHS=sin9θcos9θsin3θcos27θcos27θcos9θ+sinθcos3θ
=sin18θ+sin30θ−sin24θ2cos27θcos9θ+sinθcos3θ
=sin21θ+sin15θ+sin33θ+sin27θ−sin27θ−sin21θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ
=sin15θ+sin33θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ
=2sin24θcos9θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ
=sin24θ+2sinθcos27θ2cos3θcos27θ
=sin(27θ−3θ)2cos3θcos27θ+sinθcos3θ
=12[sin27θcos27θ−sin3θcos3θ]+sinθcos3θ
=12sin27θcos27θ−12(sin3θ−2sinθcos3θ)
=12tan27θ−123sinθ−4sin3θ−2sinθcosθ(4cos2θ−3)
=12tan27θ−tanθ
So,k=12