I meet some problems in solving some equations. I tried a lot of methods, but I didn't get any solution, so I'd be grateful if you could give some ideas.

1) , for

At first, I tried to write




but I don't know how to solve next.

Then, I wrote

If



If \cos x \neq 0 \Rightarrow \frac{\sin^3 x}{\cos^3 x} + \frac{\cos^3 }{\cos^3 x} - \frac{\cos^2 x}{\cos^3 x} + \frac{\sin^2 x}{\cos^3 x} = 0 \Rightarrow \tan^3 x + 1 - \frac{1}{\cos x} + \tan^2 \frac{1}{cos x} = 0 \Rightarrow \tan^3 x + 1 - \frac{1}{cos x} (\tan^2 x - 1) = 0 \Rightarrow (\tan x + 1)(\tan^2 x - \tan x + 1) - \frac{1}{\cos x} (\tan x -1)(\tan x + 1) = 0 \Rightarrow (\tan x +1)[\tan^2 x - \tan x +1 - \frac{1}{\cos x}(\tan x -1)] = 0 \Rightarrow (\tan x +1)[\tan x (\tan x -1) + 1 - \frac{1}{\cos x}(\tan x - 1)] = 0 \Rightarrow (\tan x +1)[(\tan x -1)(\tan x - \frac{1}{\cos x})+1] =0 \Rightarrow (\tan^2 -1)(\tan x - \frac{1}{\cos x} + \tan x + 1) = 0 \Rightarrow (\tan^x -1)(2 \tan x - frac{1}{\cos x} + 1) = 0[/math]






Is this correct? If it is, how to solve next?


2)







What to do now to get the solution?


Thank you for taking time to read this.

Try this forteh right hand side of the original equation:


You already have the left hand side as


So you can divide through by ; however, you can do this only if . Let's hold that thought for later:



Now you can divide through by , but only if :



So that's one solution. Now go back and consider the cases that we threw out earlier; namely


and


This gives you two more solutions, which I'll leave to you to solve.

Thank you ebaines! It is easier than thought. I should have kept [math]sin^2 x + cos^2 x[math] on the left side instead of writing 1.