How to we transform $(f \circ g)(x)$ into a single function, for instance: $\tan(\arccos(\frac{1}{x}))$, where the functions are from different families?
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1$$\tan \left(\arccos\left(\frac{1}{x}\right)\right)=x,\sqrt{1-\frac{1}{x^2}} $$ – Raffaele Aug 31 '17 at 11:57
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@Raffaele makes a very good point: trig-arctrig compositions—and in general compositions of mutually inverse functions—are a very different story than the composition of two otherwise unrelated functions. José Carlos Santos’s answer assumes that we don’t know anything about what function is an inverse of which (that’s what he means by “in general,” I believe). – gen-ℤ ready to perish Aug 31 '17 at 13:15
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@ChaseRyanTaylor In general there is no way to express $f(g(x))$ in a nice way, but the OP choose a (rare) example where it is possible so I wrote it down. $\cos\arctan x=\frac{1}{\sqrt{x^2+1}}$ is another nice one :) – Raffaele Aug 31 '17 at 13:31
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In general, we don't. For instance, there is no way of simplifying $\exp(\sin x)$.
José Carlos Santos
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What if it is related to another similar expression like tan(arc cosine(1/x))= sin(arc Cotangent(1/x)) – Pratyush Dutta Aug 31 '17 at 11:11
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@PratyushDutta That's too vague. There is no general answer for that. – José Carlos Santos Aug 31 '17 at 11:15
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@Pratyush If your composition has a trig function on the outside and an inverse trig function on the inside, this should be possible using, for example, a labeled right triangle. – pjs36 Aug 31 '17 at 11:20
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Solving the above equation (using a right angled triangle) we get ((x^2)-1)^(1/2)=(x/((1+x^2) ^(1/2))). From which we get x=(1+(5)^(1/2))/2,(1-(5)^(1/2))/2. However, the answer is 3/(5)^(1/2). – Pratyush Dutta Aug 31 '17 at 11:36