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.