Number of real solution of Trigonometric Equation

Question

Number of solution of the equation $(\sin x+\cos x+2)^{4}=128\sin(2x)\;\forall x\in\bigg[0,\frac{\pi}{2}\bigg]$

What i try

$$\sin x+\cos x+2=\sqrt{2}\cos\bigg(x-\frac{\pi}{4}\bigg)+2$$

And put $\displaystyle x-\frac{\pi}{4}=t$ and $\displaystyle t\in\bigg[-\frac{\pi}{4},\frac{\pi}{4}\bigg]$

$$\bigg(\cos t+\sqrt{2}\bigg)^4=-64\cos(2t)$$

How do i solve it Help me

Answer 1

From arithmetic Geometric Inequality

$$\frac{\sin x+\cos x+2}{4}\geq \bigg[\sin x\cdot \cos x\bigg]^{\frac{1}{4}}$$

$$(\sin x+\cos x+2)^4\geq 128\cdot \sin 2x$$

Equality hold when $\sin x=\cos x=1=1$

Answer 2

Let $\sin{x}+\cos{x}=t$.

Thus, by C-S $$|t|\leq\sqrt{(1+1)(\sin^2x+\cos^2x)}=\sqrt2$$ and we need to solve that $f(t)=0,$ where $$f(t)=(t+2)^4-128(t^2-1).$$ But $$f''(t)=12(t+2)^2-256\leq12(\sqrt2+2)^2-256<0,$$ which says that $f$ is a concave function on $[0,\sqrt2]$ and since $f(0)>0$ and $f(\sqrt2)>0$,

we see that our equation has no real roots.