From triangle ABC,

From Pythagoras Theorem,

     AC² = AB² + BC²

=> AC² = 3² + 4²

=> AC² = 9 + 16

=> AC² = 25

=> AC = √25

=> AC = 5

Now, cos θ = 4/5

and sin θ = 3/5

Now,

cos² 2θ - sin² 2θ = 1 - sin² 2θ  - sin² 2θ      {since cos² θ + sin² θ = 1}

                        = 1 - 2sin² 2θ

                        = 1 - 2(sin 2θ)²

                        = 1 - 2(2*sin θ *cos θ)²  

                        = 1 - 2(2*3/5 *4/5)²

                        = 1 - 2{(2*3*4)/(5*5)}²

                        = 1 - 2{24/25}²

                        = 1 - 2{576/625}

                        = 1 - 1152/625

                        = (625 - 1152)/625

                        = -527/625