Hey everyone. In this problem, we're asked to simplify the expression, and the expression we're given is the sine squared of theta minus the tangent squared of theta over the sine of theta plus the tangent of theta. Now remember, in order to fully simplify your equation, we want all of our arguments to be positive, which here they already are. And we also don't want any fractions, which we do have one big fraction here, so we want to get rid of that first. Now looking at my strategies here, we are constantly scanning for identities, but I don't really see any that will help me yet.

But one of my other strategies is going to be to factor. Looking at the numerator that I have here, sine squared of theta minus the tangent squared of theta, you may recognize this as being a difference of squares. Because this is one term squared minus another term squared, so this will factor into that first term plus that second term times that first term minus that second term. Factoring that out, I can get the sine of theta, that first term, plus the tangent of theta, second term, and then multiplying that by the sine of theta minus the tangent of theta. Now my denominator stays the same here, the sine of theta plus the tangent of theta.

But here I notice that I can cancel something because this entire denominator is exactly what this first term is now in my numerator. So this fully cancels out and I've gotten rid of my fraction because now all I'm left with is the sine of theta minus the tangent of theta. So I have no fractions here. Now, the other thing that tells us our expression is fully simplified is that we have as few trigonometric functions as possible. Now here I have 2 trigonometric functions.

However, because they're being subtracted, it's going to be kind of hard to make it so that there are any fewer trigonometrical functions than there already are. So having these 2 trigonometrical functions is as few trigonometrical functions as I can have possibly. We have also satisfied that last piece of criteria there, and we have fully simplified our trigonometric expression down to the sine of theta minus the tangent of theta. Thanks for watching, and let me know if you have any questions.

sin⁡θ2 - tan⁡θ2 sin⁡θ + tan⁡θ