If alpha = pi/13 then the value of prod_(r=1)^6 cos(ralpha) is equal to ?

If alpha = pi/13 then the value of prod_(r=1)^6 cos(ralpha) is equal to :

(A)1/64

(B)-1/64

(C)1/32

(D)-1/8

1 Answer
Jun 28, 2018

Given alpha=pi/13=>13alpha=pi

Let

P=prod_(r=1)^6 cos(ralpha)

=>P sinalpha=1/2*2sinalphacosalphaprod_(r=2)^6 cos(ralpha)

=1/4*2sin2alphacos2alphaprod_(r=3)^6 cos(ralpha)

=1/8*2sin4alphacos4alpha*cos3alpha *cos5alpha*cos6alpha

=1/8*sin8alpha*cos3alpha *cos5alpha*cos6alpha

=1/8*sin(13alpha-5alpha)*cos3alpha *cos5alpha*cos6alpha
=1/8*sin(pi-5alpha)*cos3alpha *cos5alpha*cos6alpha

=1/16*2sin5alphacos5alpha *cos3alpha*cos6alpha

=1/16*2sin5alphacos5alpha *cos3alpha*cos6alpha
=1/16sin10alpha *cos3alpha*cos6alpha

=1/16sin(13alpha -3alpha)cos3alpha*cos6alpha

=1/32*2sin(pi -3alpha)cos3alpha*cos6alpha

=1/32*2sin 3alphacos3alpha*cos6alpha

=1/64*2sin 6alphacos6alpha

=1/64*sin 12alpha

=>P sinalpha=1/64*sin (13alpha-alpha)

=>P sinalpha=1/64*sin (pi-alpha)
=>P sinalpha=1/64*sin alpha
=>P=1/64