f1  :=  c1^2  +  s1^2  -  1

f2  :=  c2^2  +  s2^2  -  1

f3  :=  l3 * (c1*c2  -  s1*s2)  +  l2*c1  -  a

f4  :=  
      l3 * (c1*s2  +  c2*s1)  +  l2 *s1  -  b

G  :=  
      GroebnerBasis[{f1, f2, f3, f4}, {c2, s2, c1, s1},  
        CoefficientDomain  ->  RationalFunctions]

Length[G]

G[[1]]

G[[2]]

G[[3]]

G[[4]]

l2  :=  1

l3  :=  1

G2 := 
      GroebnerBasis[{f1, f2, f3, f4}, {c2, s2, c1, s1}, 
        CoefficientDomain -> RationalFunctions]

Length[G2]

G2

G

Solve[G[[1]] == 0 ,  s1 ]

a :=  0

G3 := 
      GroebnerBasis[{f1, f2, f3, f4}, {c2, s2, c1, s1}, 
        CoefficientDomain -> RationalFunctions]

G3

b :=  0

G4 := 
      GroebnerBasis[{f1, f2, f3, f4}, {c2, s2, c1, s1}, 
        CoefficientDomain -> RationalFunctions]

G4