Find the values of $a$ for which the function $f(x)=\sin (2x)-8(a+1) \sin x+(4a^2+8a-14)x$ increases for all $x\in R$. 
Find the values of $a$ for which the function $f(x)=\sin (2x)-8(a+1) \sin x+(4a^2+8a-14)x$ increases for all $x\in R$ and has no critical points.

For increasing function, $$f'(x)\gt0\\2\cos(2x) - 8(a+1)\cos x + (4a^2 + 8a -14)\gt0$$
Using $\cos(2x)=2\cos^2x-1$, $$4\cos^2x-8(a+1)\cos x+4a^2+8a-16\gt0\\\cos^2x-2(a+1)\cos x+a^2+2a-4\gt0\\(\cos x-(a+1))^2-5\gt0\\(\cos x-a-1+\sqrt5)(\cos x-a-1-\sqrt5)\gt0\\\cos x\lt a+1-\sqrt5 \text{ or }\cos x\gt a+1+\sqrt5$$
Now, I know that $\cos x$ lies between $-1$ and $1$. Not able to use that to solve the inequality.
Also, the same question has been discussed here, but the approach is different. Also, the final answer posted there is not correct.

By your work we need $$\cos^2x-2(a+1)\cos x+a^2+2a-4>0.$$
We have three cases:

*

*$a+1>0$. In this case we need $$1^2-2(a+1)\cdot1+a^2+2a-4>0.$$

*$a+1<-1$. In this case we need $$(-1)^2-2(a+1)\cdot(-1)+a^2+2a-4>0.$$

*$$(a+1)^2-(a^2+2a-4)<0.$$
Can you end it now?

