• # question_answer The graph of the function y = f[x} = cos x cos [x + 2) - cos2 [x + 1) is A)  a straight line through $(0,-{{\sin }^{2}}1)$ with slope 2 B)  a straight line through (0, 0) C) a parabola with vertex $(1,-{{\sin }^{2}}1)$ D)  a straight line through $\left( \frac{\pi }{2},-{{\sin }^{2}}1 \right)$ and parallel to x-axis

We have given that $f(x)=\cos x\cos (x+2)-{{\cos }^{2}}(x+1)$ $=\cos (x+1-1)\cos (x+1+1)-{{\cos }^{2}}(x+1)$ $={{\cos }^{2}}(x+1)-{{\sin }^{2}}1-{{\cos }^{2}}(x+1)$ $\because$$\cos (A+B)\cos (A-B)={{\cos }^{2}}A-{{\sin }^{2}}B$                                 $f(x)=-{{\sin }^{2}}1$ i.e.,            $y=-{{\sin }^{2}}1$ (a constant quantity) Hence, the graph is a straight line parallel to x-axis and passing through $\left( \frac{\pi }{2},-{{\sin }^{2}}1 \right).$ Here, $\frac{\pi }{2}$ can be replaced by any real number.