Find the minimum value of $(\sin x+ \csc x)^2 + (\sec x + \cos x)^2$ I made it till $5+\tan^2x + \cot^2x$, but I don’t know how to proceed further.

I don't think your function is $5+\tan^2x+\cot^2x$, actually. With $c:=\cos x,\,s:=\sin x$ your sum is $$\left(s+\frac{1}{s}\right)^2+\left(c+\frac{1}{c}\right)^2=5+\frac{1}{s^2}+\frac{1}{c^2}=5+\frac{1}{c^2s^2}=5+4\csc^22x\ge 9,$$with equality iff $\sin2x=\pm 1$, as occurs e.g. if $x=\frac{\pi}{4}$ for which we get$$\left(s+\frac{1}{s}\right)^2+\left(c+\frac{1}{c}\right)^2=2\left(\frac{1}{\sqrt{2}}+\sqrt{2}\right)^2=2\left(\frac{3}{\sqrt{2}}\right)^2=9.$$
