Okay I now see that you can rewrite the two trig functions in terms of sine and cosine but what now?
Given the standard hyperbola I need to find the area bounded by the curve and the line
Here is the visual representation
Solving for y gives me
I then set up the integral
I set , substituted and changed limits of integration accordingly
Using a trig identity I can simplify down to
And that is where I am stuck, I tried another u-substitution to no avail.
I think your integral wont work because half the area is above the x axis and have the area is below the x axis (negative) so the integral will be 0.
I would integrate with respect to y instead
Solving the equation of the hyperbola for x gives
To find the area between two lines you integrate the difference between them.
The limits of the integration have an x coordinate 2a and satisfy the equation of the hyperbola, they turn out to be
The OP's integral is correct, the integral of the function between a and 2a would give the area above the x-axis, doubling that gives the total area.
It would be easier to write the integral as , then you can make the substitution , noting that when and when and your integral becomes