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#### abender

##### New member

- Jan 8, 2013

- 4

Please help me to find the range of [TEX]sec^{4}(x)+cosec^{4}(x)[/TEX].

Thanks in advance.

[TEX]\sec^4(x) = (\sec^2(x))^2 = (1+\tan^2(x))^2 = \tan^4(x) + 2\tan^2 + 1[/TEX]

[TEX]\csc^4(x) = (\csc^2(x))^2 = (1+\cot^2(x))^2 = \cot^4(x) + 2\cot^2 + 1 [/TEX]

[TEX]\sec^4(x) + \csc^4(x) = \tan^4(x) + 2\tan^2 + 1 + \cot^4(x) + 2\cot^2 + 1 [/TEX][tex] = \tan^4(x) + \cot^4(x) + 2(\tan^2(x)+\cot^2(x)) + 2[/tex]

The range of [TEX]\tan(x)[/TEX] is [TEX](-\infty, \infty)[/TEX]. Likewise, the range of [TEX]\cot(x)[/TEX] is [TEX](-\infty, \infty)[/TEX].

Since [TEX]\tan^4(x)[/TEX] and [TEX]\cot^4(x)[/TEX] are positive even powers, both have range [TEX][0,\infty)[/TEX].

BELOW is where others may disagree with me:

I contend that the range of [TEX]\tan^4(x)+\cot^4(x)[/TEX] is [TEX](0,\infty)[/TEX] as opposed to [TEX][0,\infty)[/TEX], which the sum of the parts may intuitively suggest.

The two ranges differ insofar [TEX](0,\infty)[/TEX]contain [TEX]0[/TEX], whereas [TEX][0,\infty)[/TEX]does notcontain [TEX]0[/TEX].does

I believe that the range of [TEX]\tan^4(x)+\cot^4(x)[/TEX] should NOT include 0, i.e., it should be [TEX](0,\infty)[/TEX].

Proof is achieved if we show [TEX]\tan^4(x)+\cot^4(x)>0[/TEX] on the entire domain (reals that are not multiples of [TEX]\pi[/TEX]).

Both terms in [TEX]\tan^4(x)+\cot^4(x)[/TEX] are non-negative in the reals, so clearly the sum itself is non-negative.

[TEX]\tan^4(x)+\cot^4(x)[/TEX] cannot equal 0 because then either [TEX]\tan^4(x)=-\cot^4(x)[/TEX] (which per the line above is not possible) OR [TEX]\tan^4(x)=\cot^4(x)=0[/TEX], which also can't happen because [TEX]\cot^4(x)=\tfrac{1}{\tan^4(x)}[/TEX] and 0 cannot be a denominator.

As such, [TEX]\tan^4(x)+\cot^4(x)>0[/TEX].

And with this new information, [TEX]\sec^4(x) + \csc^4(x) = \underbrace{\left[\tan^4(x) + \cot^4(x)\right]}_{\text{always positive!}} + 2\underbrace{(\tan^2(x)+\cot^2(x))}_{\text{can show positive similarly}} + 2 > 2[/TEX].

It at long last follows that the range of [TEX]\sec^4(x) + \csc^4(x)[/TEX] is [TEX]\left(2,\infty\right)[/TEX].

If someone has an argument for a left bracket instead of my left open parenthesis, I'd love to hear it!

-Andy