Given the function:
First, the graph of the function will be as shown in the following figure:
For the given rational function, we will find the following:
Domain = ( -∞, 0 ) ∪ ( 0, ∞ )
Range = ( -∞, 2) ∪ (2, ∞ )
Increasing = Φ
Decreasing = ( -∞, 2) ∪ (2, ∞ )
All asymptotes:
Vertical asymptote: x = 0
Horizontal Asymptote: y = 2
All limits:
Answer:
(the relation you wrote is not correct, there may be something missing, so I will simplify the initial expression)
Here we have the equation:
We can rewrite this as:
Now we can add and subtract cos^2(x)*sin^2(x) to get:
We can complete squares to get:
and we know that:
cos^2(x) + sin^2(x) = 1
then:
This is the closest expression to what you wrote.
We also know that:
sin(x)*cos(x) = (1/2)*sin(2*x)
If we replace that, we get:
Then the simplification is: