Simplifying Compositions of Trigonometric Functions and Inverse Trigonometric Functions

The composition of a function and its own inverse should get you back where you started. This pretty much works with trigonometric functions in spite of the fact that the inverses aren't really true inverses.

sin(arcsinx)=x
cos(arccosx)=x
tan(arctanx)=x
cot(arccotx)=x
csc(arccscx)=x
sec(arcsecx)=x

If you reverse the order and talk about such things as arcsin(sinx), you have to be a bit more careful, because of the restricted ranges of the inverse functions, but for now I would like to stick to the first order, but make things a little more complicated by making the original and the inverses different trigonometric functions, like for example cos(arcsinx). Because of the various identities relating the trigonometric function there are always nice simplifications for things like this. One way you could simplify cos(arcsinx) would be by writing cosine in terms of sine by using the Pythagorean Identities. If you did that you would get

(Actually this isn't quite legal because it should be plus or minus, but because of the way the restricted ranges were defined it works anyway, and in fact I think some care was actually taken when defining all of the restricted ranges so that it always works to throw away the plus or minus in these sorts of problems. For example in this case, the arcsin always gives you a value that is in the first or fourth quadrant, which is exactly where the cosine is positive. In any case, whether by design or mere chance, it turns out that in all these problems you don't have to worry about the plus or minus in the radicals because of the ways that the ranges were chosen.)

So anyway, you can use an identity to simplify this to an expression that doesn't even have anything to do with trigonometry. And if you think about it, you realize that since you can write any trig function in terms of the others you ought to be able to always do something like this. But if you have something more complicated than cos(arcsinx), like maybe tan(arccos), then it can get sort of complicated with the identities, so I'll show you a simpler way to do it.

The simpler way involves triangles, so it is sort of cheating, because that means you are pretending all of your angles are in the first quadrant. But it works anyway, again because of the way the ranges of the inverse functions were defined. So here's what you do.

To evaluate cos(arcsinx) you simply draw arcsinx on a triangle and then find its cosine.

By putting x on the opposite side and 1 on the hypotenuse you force the angle to have a sine of x and that makes it be the arcsin of x. Then to compute its cosine you need to find the adjacent side, but you can use the Pythagorean Theorem for that. Then once you know the adjacent side, you can compute the cosine by cos=adj/hyp and get the same simplification we got before. You can use this method for any combination of trig function and inverse trig function quite easily, so for example if you wanted tan(arccosx), you would make the adjacent side x and the hypotenuse 1. Then in order to find the tangent you would need the opposite side, but you could get that by using the Pythagorean Theorem, and then you would divide the opposite by the adjacent to get the tangent. This method works well with numbers too, especially ones that give you nice Pythagorean Triple triangles like 3,4,5. To find sec(arctan(3/4)) just put 3 on the opposite side and 4 on the adjacent side and then the hypotenuse will be 5 and you can compute the secant of the angle and you get 5/4, so you can do it without ever figuring out what arctan(3/4) is.