Solve Trig Identity Problem: Eliminating θ from x=2+cscθ and y=1/4tanθ

[Briggs]

I have been set an exercise from my textbook to do for homework but I am having some problems on one of the questions, I havn't encountered any of this type before and I am quite stumped.

"Eliminate $$\theta$$ from equations $$x=2+\csc\theta$$ and $$y=\frac{1}{4}\tan\theta$$"

So I am guessing I start off with x=y but then...

 

[Fermat]

start off by getting separate expressions for cscθ and tanθ from each of your two eqns.
Now you have to eliminate θ.
Can you do it from there ?

 

[Briggs]

So for the first one would it goto,
$$2+\frac{1}{\sin\theta}=3+\cot\theta$$ Then try eliminate theta from there?

 

[Fermat]

not quite.
get two separate expressions. One for cscθ and one for tanθ.
Then find a trig identity involving the trig functions. You will have to manipulate the functions to get the identity.

 

[Briggs]

I'm not sure what you are saying, get an expression for each equation so for the first one would become $$2+\cot\theta$$ and the second one $$\frac{1}{4}\csc\theta-\frac{1}{4}$$ then put them equal to each other and try solve it?
I would appreciate it if you could show me the first step in tex form

 

[Fermat]

hang on a minute.

 

[Fermat]

x = 2 + cscθ
y = ¼tanθ

cscθ = x - 2
tanθ = 4y

1 + tan²θ = sec²θ

 

[Briggs]

Ahhh, it all becomes clear. Thanks a lot for the help It seems so easy now.