What is wrong in this solution of $\sec x + \csc x = 2 \sqrt{2}$ 
Find number of solutions in the interval $[0,2\pi]$ of the equation - $$\csc x + \sec x = 2 \sqrt{2}.$$

$⇒ \dfrac{1}{\sin x}+ \dfrac{1}{\cos x }= 2 \sqrt2$
$⇒ \dfrac{\sin x +\cos x}{\sin x  \cos x} = 2 \sqrt 2 ⇒ (\sin x+\cos x)^2=8\sin^2x\cos^2x⇒1+2\sin x \cos x=8\sin^2x \cos^2x$
$⇒\sin x \cos x=-1/4 ,\sin x \cos x=1/2⇒2\sin x \cos x=-1/2 , 2\sin x\cos x=1$
$⇒\sin2x =-1/2 , \sin2x=1.$ But if we check manually , $\sin2x=-1/2$ is not a solution.
$⇒\sin 2x =1⇒2x=(-1)^{n}\dfrac{\pi}{2}+n\pi⇒x=(-1)^{n}\dfrac{\pi}{4}+\dfrac{n\pi}{2}⇒$ For $[0,2\pi], \boxed{x=\dfrac{\pi}{4},\dfrac{5 \pi}{4}}$ are solutions.
But if we check the graph neither are the solutions . But $\pi /4 $ is a solution but $5\pi /4 $ is not a solution if we plug values manually.

Can someone point out the mistake? Thanks
$\text{References:}$ Link for above graph : https://www.desmos.com/calculator/eaakqvd9y5

Hint:
Instead of squaring $$\dfrac {\sin x + \cos x}{\sin x \cos x} = 2\sqrt{2}$$ clear fractions to get $$\sin x + \cos x = 2\sqrt{2}\sin x \cos x$$ and note that you have a double angle identity hidden on the RHS.  Then you can use another identity to get the LHS and find the relationship between the two.
