if 1/6 sinx ,cosx , tanx , are in GP then find the general solution of x

Hello Student,

1/6 sinx, cosx and tanx are in G.P. So
cos^2x=1/6sinx.tanx
6cos^3x=sin^2x=1-cos^2x
6cos^3x+ cos^2x-1=0
put cosx=y then
6y^3+y^2-1=0
put y=1/2
6/8 +1/4-1=3/4+1/4-1=1-1=0
hence ( y-1/2) is the factor of above eq.
hence the factors are
(y-1/2)(6y^2+4y+2)=0
=>(y-1/2)=0 and second factor have imaginary roots.
so y=1/2
cosx=1/2=cospi/3
x=pi/3

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty