Minimum value of this particular trigonomterical function : really interesting

First take a look at the problem :

Find the minimum value of the trigonometrical function :

If we proceed by perfect-square method , we get ,

=>

=>

Therefore ,

But this minimum value will be achieved if and only if ,

which when solved gives ( approximately ) , which is certainly not possible since maximum value of is 1

Does this mean that the function does not have any minimum value .

But if we plot the graph of , it looks like it has a minimum value of 13 .

Please analyze and explain !