1)

when  sin9θcos27θ+sin3θcos9θ+sinθcos3θ=k

 (tan27θtanθ)  is defined , then k=


A) π2

B) -12

C) 12

D) π4

Answer:

Option C

Explanation:

sin9θcos27θ+sin3θcos9θ+sinθcos3θ=k(tan27θtanθ) 

LHS=sin9θcos9θsin3θcos27θcos27θcos9θ+sinθcos3θ

=sin18θ+sin30θsin24θ2cos27θcos9θ+sinθcos3θ

=sin21θ+sin15θ+sin33θ+sin27θsin27θsin21θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ

=sin15θ+sin33θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ

=2sin24θcos9θ+4sinθcos9θcos27θ4cos3θcos9θcos27θ

=sin24θ+2sinθcos27θ2cos3θcos27θ

=sin(27θ3θ)2cos3θcos27θ+sinθcos3θ

=12[sin27θcos27θsin3θcos3θ]+sinθcos3θ

=12sin27θcos27θ12(sin3θ2sinθcos3θ)

=12tan27θ123sinθ4sin3θ2sinθcosθ(4cos2θ3)

=12tan27θtanθ

So,k=12