(* #4, 5, 14, p. 528 *)
Solve[Sin[x]^2==4-2Cos[x]^2,x]

Solve[1-Cos[x]^2==4-2Cos[x]^2,x]

Solve[3==Cos[x]^2,x]
(* at this point it should be clear that there are "no solutions": Mathematica is finding complex (not real) solutions: *)

(* #5 *)
Solve[Tan[x]^2-2/Cos[x]==2,x]

Solve[Sin[x]^2-2*Cos[x]==2*Cos[x]^2,x]

Solve[1-Cos[x]^2-2*Cos[x]==2*Cos[x]^2,x]

Solve[1-3Cos[x]^2-2*Cos[x]==0,x]

Solve[(1-3Cos[x])(1+Cos[x])==0,x]

(* #14, p. 528 *)
Solve[Cos[x]-Sin[x]==1,x]

Solve[(Cos[x]-Sin[x])^2==1,x]

Solve[Cos[x]^2-2Cos[x]Sin[x]+Sin[x]^2==1,x]

Solve[-2Cos[x]Sin[x]==0,x]