FINALLY

Finally, after hours of work, I have figured out the secret to trigonometric substitutions.

The biggest tip I can give is to restrict the domain of your sinθ, secθ, and tanθ functions.

For sinθ -π/2 < θ < π/2

      secθ  0 < θ < π/2 OR π < θ < 3π/2
    
      tanθ -π/2 < θ < π/2


NOTE:  Be cautious when it comes to reference triangles.  Most of these will only need one reference triangle due to the restriction of the domain.