MHB Solving a Trigonometric Equation

AI Thread Summary
The equation tan^4(x) + tan^2(x) = sec^4(x) - sec^2(x) simplifies to an identity, meaning it holds true for all x in the domain. By rearranging and factoring, it can be shown that both sides are equal, confirming the identity. The manipulation involves using the relationship between tangent and secant, specifically that tan^2(x) + 1 = sec^2(x). Various approaches to solving or proving the identity were discussed, all leading to the same conclusion. The consensus is that the original equation is indeed an identity rather than a solvable equation.
thorpelizts
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solve for

tan^4x + tan^2x = sec ^4x - sec^2x

i solved and ended up with RIHS= tan^4x?
 
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re: Solving a Trignometric Equation

We are given to solve:

$\displaystyle \tan^4(x)+\tan^2(x)=\sec^4(x)-\sec^2(x)$

I would arrange as:

$\displaystyle \tan^4(x)-\sec^4(x)+\tan^2(x)+\sec^2(x)=0$

Factor:

$\displaystyle (\tan^2(x)+\sec^2(x))(\tan^2(x)-\sec^2(x))+\tan^2(x)+\sec^2(x)=0$

$\displaystyle (\tan^2(x)+\sec^2(x))((\tan^2(x)-\sec^2(x))+1)=0$

Now, since $\displaystyle \tan^2(x)+1=\sec^2(x)$ we have:

$\displaystyle 0=0$

which means the original equation is an identity, i.e., it is true for all values of x in the domain.

Were you supposed to prove the identity is true instead of solving the equation?
 
re: Solving a Trignometric Equation

yeah, thx
 
re: Solving a Trignometric Equation

thorpelizts said:
solve for

tan^4x + tan^2x = sec ^4x - sec^2x

i solved and ended up with RIHS= tan^4x?

If You apply the basic definitions the 'equation' becomes...

$\displaystyle \frac{\sin^{4} x}{\cos^{4} x} + \frac{\sin^{2} x}{\cos^{2} x} = \frac{1}{\cos^{4} x} - \frac{1}{\cos^{2} x} \implies \frac{\sin^{4} x-1}{\cos^{4} x} + \frac{\sin^{2} x+1}{\cos^{2} x}=0 \implies$

$\displaystyle \implies \frac{\sin^{2} x -1+ \cos^{2} x}{\cos^{4} x} =0 \implies \frac{0}{\cos^{4} x}=0$

... anf that is an identity, i.e. any x satisfies the 'equation'...

Kind regards

$\chi$ $\sigma$
 
re: Solving a Trignometric Equation

I like to begin with the left side, and try to manipulate it so that the right side results. I think I would first factor the left side to get:

$\displaystyle \tan^2(x)(\tan^2(x)+1)$

Now, use the Pythagorean identity $\displaystyle \tan^2(x)+1=\sec^2(x)$ and see where this leads you...
 
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