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no difference, both are negatives.

But, I don't understand this:tan (-alpha) = -tan alpha

I don't understand.

what exactly is your doubt ?

so (tan (-x) is from the 4th quadrant and -tan x in the first, how can it be negative in the first.

What is a ratio?

sin,cos , tan,.... are ratios...

I a making an ex: and see this: tan x = -1 <=> - tan x (pi/4)
Why is the - before tan?