• question_answer Number of possible ordered pair(s) (a, b) for each of which the equality, $a\,(\cos x-1)+{{b}^{2}}=\cos \,(ax+{{b}^{2}})-1$ holds true for all $x\in R$ are: A) 0 B) 1 C) 2 D)  infinite

 Put $x=0$ ${{b}^{2}}=\cos$ ${{b}^{2}}-1$ or $\cos {{b}^{2}}=1+{{b}^{2}}$$\Rightarrow$$b=0$ Now the equation becomes $-\,2\,\,a{{\sin }^{2}}\frac{x}{2}=\cos a\,x-1=-\,2{{\sin }^{2}}\frac{ax}{2}$ or $a{{\sin }^{2}}\frac{x}{2}={{\sin }^{2}}\frac{ax}{2}$ must be true $\forall \,\,x\in R$$\Rightarrow$$a=0$ or $a=1$