How do I simplify this pure trigonometric equation? sec^4(x) = tan^4(x) + 3tan(x)

[Al-Layth]

So I'm trying to solve this trig equation:

[math]\sec^{4}(x)=\tan^4{x}+3\tan(x)[/math]
First I tried to convert everything into sin, cos and see if I could solve the resultant equation(this is what I always do, it works 100% of the time, 70% of the time). I got:
[math]1=\sin^4(x)+3\sin(x)\cos^3(x)[/math]
Which I couldn't solve(if you know how to solve that then please tell me)
. I put the original equation into an online computer algebra system, and it told me the original equation is equivalent to:

[math]2\sec^2(x)=3\tan(x)+1[/math]Now this is very easily solvable. The issue is, how does a person get from the original equation to this simplified version?
And in this case, the number of terms containing trig has been reduced by one, and the trig powers have been reduced too. This is like a perfect example of simplification, and how does a person determine whether something like this is possible given a trig equation.

Also an extra question: If I were to rewrite a trig equation in just sin and cos, and then attempt to simplify using only the identities relevant to sin and cos, would it be equivalent to trying to simplify the original trig equation using the full body of trig identities? Or would I "miss out" on some opportunity to simplify if i converted the equation into a function of sin and cos first.

 

[Steven G]

[math]2\sec^2(x)=3\tan(x)+1[/math]

That is the new problem that you need to show:
Show that sec⁴(x) - tan⁴(x)= 2sec²(x) - 1

 

[Dr.Peterson]

[math]\sec^{4}(x)=\tan^4{x}+3\tan(x)[/math]
I put the original equation into an online computer algebra system, and it told me the original equation is equivalent to:

[math]2\sec^2(x)=3\tan(x)+1[/math]The issue is, how does a person get from the original equation to this simplified version?
And in this case, the number of terms containing trig has been reduced by one, and the trig powers have been reduced too. This is like a perfect example of simplification, and how does a person determine whether something like this is possible given a trig equation.

I see a difference of 4th powers, which can be factored:
[math]\sec^{4}(x)-\tan^4{x}=3\tan(x)[/math][math](\sec^{2}(x)-\tan^2{x})(\sec^{2}(x)+\tan^2{x})=3\tan(x)[/math][math](1)(\sec^{2}(x)+\tan^2{x})=3\tan(x)[/math][math]\sec^{2}(x)+(\sec^2{x}-1)=3\tan(x)[/math][math]2\sec^{2}(x)-1=3\tan(x)[/math][math]2\sec^{2}(x)=3\tan(x)+1[/math]
You determine that it's possible by trying it ...

Also an extra question: If I were to rewrite a trig equation in just sin and cos, and then attempt to simplify using only the identities relevant to sin and cos, would it be equivalent to trying to simplify the original trig equation using the full body of trig identities? Or would I "miss out" on some opportunity to simplify if i converted the equation into a function of sin and cos first.

Yes, I think it's equivalent to using all the identities.

 

[khansaheb]

So I'm trying to solve this trig equation:

[math]\sec^{4}(x)=\tan^4{x}+3\tan(x)[/math]
First I tried to convert everything into sin, cos and see if I could solve the resultant equation(this is what I always do, it works 100% of the time, 70% of the time). I got:
[math]1=\sin^4(x)+3\sin(x)\cos^3(x)[/math]
Which I couldn't solve(if you know how to solve that then please tell me)
. I put the original equation into an online computer algebra system, and it told me the original equation is equivalent to:

[math]2\sec^2(x)=3\tan(x)+1[/math]Now this is very easily solvable. The issue is, how does a person get from the original equation to this simplified version?
And in this case, the number of terms containing trig has been reduced by one, and the trig powers have been reduced too. This is like a perfect example of simplification, and how does a person determine whether something like this is possible given a trig equation.

Also an extra question: If I were to rewrite a trig equation in just sin and cos, and then attempt to simplify using only the identities relevant to sin and cos, would it be equivalent to trying to simplify the original trig equation using the full body of trig identities? Or would I "miss out" on some opportunity to simplify if i converted the equation into a function of sin and cos first.

Use

sec²(x) = tan²(x) + 1 ............ within proper domain