# Finding Unknown Function f(x)

1. Jan 29, 2010

### fled143

1. The problem statement, all variables and given/known data

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)

2. Relevant equations

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

3. 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(*)

Last edited: Jan 29, 2010
2. Jan 29, 2010

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

3. Jan 29, 2010

### 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.

4. Jan 29, 2010

### Staff: Mentor

The problem statement is somewhat ambiguous.
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).