Plot[(1+3 Cos[q]^2),{q,0,Pi}]

Plot[(3/5) (15/(8 Pi)) Sin[q]^2 Cos[q]^2 +
  (2/5) (5 / (4 Pi)) ((3/2) Cos[q]^2 -(1/2))^2,{q,0,Pi}]

Show[%1,%2]

Simplify[(3/5) (15/(8 Pi)) Sin[q]^2 Cos[q]^2 +
  (2/5) (5 / (4 Pi)) ((3/2) Cos[q]^2 -(1/2))^2]

Plot[(5+3 Cos[2 q])/(16 Pi),{q,0,Pi}]