Let S=x∈(-π,π):x≠0,±π2 . The sum of all distinct solutions of the equation 3secx+cosecx+2(tanx-cotx)=0 in the set S is equal to

3 secx+cosecx+2tanx−cotx=0⇒3cosx+1sinx=2cosxsinx-sinxcosx⇒3sinx+cosx=2cos2x⇒32sinx+12cosx=cos2x⇒cosx-π3=cos2x⇒32sinx+12cosx=cos2x⇒cosx-π3-cos2x=0 Use transformation and equate each term to zero sin3x-π32=0 or sinx+π32=0⇒3x-π3=2nπ,x+π3=2mπ⇒π9,7π9,-π3,-5π9 ⇒ sum =0