Quite Shocking Result

This is what i think.

Using method 2 you got a result that exists but is indeterminate.(this means that it may be an exact value but cannot be determined by just using this one expression)
Using the other method you got the exact limiting value of the same expression.
Hence it is correct to equate the two.
Just a small error tan²x/2→1 and not equal to one.
Hence the following values are the limiting value of the trig. ratios.

But if we just look just into this line
lim_{x→∞} tanx =±∞
it may seem as if it is not true and a bit absurd.
But to make this statement true we must also state that:

let L2=lim_{x→∞} tanx=lim_{x→∞} 2tanx/2
1-tan²x/2
we know L=lim_{x→∞} tan²x/2 .

We can also say that lim_{x→∞} tanx/2=±L (because L=1 )

hence L2=2*±L
1-L

Where L=lim_{x→∞} x+sinx =1
x-sinx

Now we can say L2=2*±L/(1-L)=±∞

Hence L2=lim_{x→∞} tanx=±∞

Now this line makes sense.(If and only if the bold lines are true)

Can we also do these by writing the expansion.?