How Do You Solve This Complex Trigonometric Equation?

[wngman510]

Homework Statement

Constants a,b,c,d

Homework Equations

a/((1-b*cos(x))^1) + c = cos(x+d)/(1-b*cos(x)), solve for x

The Attempt at a Solution

I've tried many different forms and substutions to try to factor this puppy and get 2 solutions. The expanded equation, after using the cos sum identity, looks like

p*cos(x)^2 + q*cos(x) + r*sin(x) + s*sin(x)*cos(x) = t

I attempted to eliminate the equation by multiplying t by sin(x)^2+cos(x)^2 and finding some common terms but so far that hasn't helped. I've also trying substituting all the half and double angle identities I know and haven't been able to come up with anything. I know this doesn't show much work, but I have spent a lot of time on this, so any help is much appreciated!

 

[Mark44]

Homework Statement

Constants a,b,c,d

Homework Equations

a/((1-b*cos(x))^1) + c = cos(x+d)/(1-b*cos(x)), solve for x

Why do you have an exponent of 1 on the left side? Did you leave something out?

The Attempt at a Solution

I've tried many different forms and substutions to try to factor this puppy and get 2 solutions. The expanded equation, after using the cos sum identity, looks like

p*cos(x)^2 + q*cos(x) + r*sin(x) + s*sin(x)*cos(x) = t

I attempted to eliminate the equation by multiplying t by sin(x)^2+cos(x)^2 and finding some common terms but so far that hasn't helped. I've also trying substituting all the half and double angle identities I know and haven't been able to come up with anything. I know this doesn't show much work, but I have spent a lot of time on this, so any help is much appreciated!

 

[wngman510]

Yes, thanks. That is a 2. Pretty significant typo :)

a/((1-b*cos(x))^2) + c = cos(x+d)/(1-b*cos(x)),

 

[haruspex]

You could manipulate it into the form sin(x) = f(cos(x)) then square both sides, substituting 1-cos²(x) on the left. I think that will give you a quartic in cos(x). Not pretty.