6.If x+y=4,xy=1, then what is the value of tan^-1x+tan^-y?

This is the math equivalent of fluttering cards in someone's face. First add spaces so things don't all blur together.

x + y = 4
xy = 1

You have two lines. They look like this: www.wolframalpha.com/input/?i=plot+x+%2B+y+%3D+4,+xy+%3D+1

Where they cross, the values of x and y satisfy both equations simultaneously, so it is called a simultaneous system. There are two ways to find that point. Substitution is when you solve one equation for one variable and substitute that into the other equation.

x + y = 4
y = 4 - x Substitute this for y in the other equation.
x(4 - x) = 1
-x^2 + 4x - 1 = 0
x^2 - 4x + 1 = 0 Standard form.

Quadratic equation time.
Negative b is the place to start,
Plus or minus will show you're smart,
The radical sits so gent-i-ly,
On b square minus 4ac
Now we're done ... except to say,
The whole thing sits on top of 2a

x = 2 ± √3 and y = 2 ± √3 but x is not equal to y.
Check: www.wolframalpha.com/input/?i=solve+x+%2B+y+%3D+4,+xy+%3D+1

Elimination is when you add the equations in a way to eliminate one variable. Otherwise the process is the same. That does not apply in this case.

The rest of the question is up to you.

When I say "fluttering cards in someone's face" I mean that math is only valid when it describes reality. It is quite a stretch to find any example of reality that involves multiplying two angles.