LHS=11+tan2A2+11+tan2B2+11+tan2C2
=cos2A2+cos2B2+cos2C2
=1+cosA2+1+cosB2+1+cosC2
=32+12(cosA+cosB+cosC)
=32+12[1−2sin2A2+2cos(B+C2)cos(B−C2)]
=32+12[1−2sin2A2−2sinA2cos(B−C2)]
=2−sinA2[sinA2−cos(B−C2)]
=2−sinA2[cos(B+C2)−cos(B−C2)]
=2+2sinA2sinB2sinC2
=2[1+sinA2sinB2sinC2]
RHS=k[1+sinA2sinB2sinC2]
∴k=2
Alternate solution:
If we assume A=B=C=60∘, we get
11+tan2A2+11+tan2B2+11+tan2C2
=cos2A2+cos2B2+cos2C2=cos230∘+cos230∘+cos230∘=3×34=94
k[1+sinA2sinB2sinC2]=k[1+(sin30∘)3]=k[1+18]=9k8
Now,
9k8=94∴k=2