How Does f(x) = 1/sin(x) Satisfy the Given Functional Equation?

[fled143]

Homework Statement

f(x) = f(x-k) f(k) / [ cot(k) + cot(x-k) ]

Show that the solution of the equation is

f(x) = 1/sin(x)

Homework Equations

sin(-x) = -sin(x)
cot(x) = cos(x) / sin(x)

The Attempt at a Solution

Transform the cotangents into cos and sin and simplify

f(x) = f(k) f(x-k) sin(k) (-csc(x) ) sin(x-k) eqn(*)

 

[Mentallic]

All you have to do is substitute $f(x)=csc(x)$ into the equation and show that both sides are equal through simplication and use of trig identities.

$$f(x)=\frac{f(x-k)f(k)}{cot(k)+cot(x-k)}$$

$f(x)=csc(x)$ and this means by its definition that $f(x-k)=csc(x-k)$ and $f(k)=csc(k)$

Now you just need to show that $$csc(x)=\frac{csc(x-k)csc(k)}{cot(k)+cot(x-k)}$$

 

[fled143]

That is supposed to be easy assuming that I already know what the f(x) is. But actually the problem is that I need to derive the solution f(x) = csc(x) from the given equation. I'm sorry if I have not pose my problem clearly at the start.

Thanks for helping.

 

[Mark44]

The problem statement is somewhat ambiguous.

Show that the solution of the equation is
f(x) = 1/sin(x)

One possible meaning for this sentence is that you need to show that the function f(x) = 1/sin(x) satisfies the given equation. In this case you are given that f(x) = 1/sin(x), which is also equal by definition to csc(x).

Another meaning that IMO is less likely is that you are supposed to solve the given equation and arrive at the solution f(x) = 1/sin(x). I don't believe that this is the intent of the problem. If that had been the case, the problem writer could have been clearer by asking you to solve the given equation for f(x).