Please help me to find the range of $\displaystyle sec^{4}(x)+cosec^{4}(x)$.

Thanks in advance.

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- Feb 3rd 2013, 05:37 AMkjchauhanRange
Please help me to find the range of $\displaystyle sec^{4}(x)+cosec^{4}(x)$.

Thanks in advance. - Feb 3rd 2013, 06:19 AMPlatoRe: Range
- Feb 3rd 2013, 10:05 AMabenderRe: Range
$\displaystyle \sec^4(x) = (\sec^2(x))^2 = (1+\tan^2(x))^2 = \tan^4(x) + 2\tan^2 + 1$

$\displaystyle \csc^4(x) = (\csc^2(x))^2 = (1+\cot^2(x))^2 = \cot^4(x) + 2\cot^2 + 1 $

$\displaystyle \sec^4(x) + \csc^4(x) = \tan^4(x) + 2\tan^2 + 1 + \cot^4(x) + 2\cot^2 + 1 $$\displaystyle = \tan^4(x) + \cot^4(x) + 2(\tan^2(x)+\cot^2(x)) + 2$

The range of $\displaystyle \tan(x)$ is $\displaystyle (-\infty, \infty)$. Likewise, the range of $\displaystyle \cot(x)$ is $\displaystyle (-\infty, \infty)$.

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

__BELOW is where others may disagree with me:__

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

The two ranges differ insofar $\displaystyle (0,\infty)$contain $\displaystyle 0$, whereas $\displaystyle [0,\infty)$**does not**contain $\displaystyle 0$.**does**

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

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

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

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

As such, $\displaystyle \tan^4(x)+\cot^4(x)>0$.

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

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

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

-Andy - Feb 3rd 2013, 10:46 AMPlatoRe: Range
- Feb 3rd 2013, 11:21 AMabenderRe: Range
$\displaystyle [8,\infty)$, evidently. My way was a fun adventure; I wanted to do it using identities for some bizarre reason. These problems nearly always boil down to the rudimentary sines and cosines. I see the airtight solution now, the symmetry, no plus 2 times "something positive". In retrospect, why the hell did I accept 2 as my best lower bound? Oh well, my Ravens are in the Super Bowl! Usually never post if I'm uncertain.