# Thread: Express the following by using 't'

1. ## Express the following by using 't'

When

$t=\frac{{a^{2}+b^{2}}}{{a^{2}-b^{2}}}+
\frac{{a^{2}-b^{2}}}{{a^{2}+b^{2}}}$

express the following by using 't'

$\frac{{a^{8}+b^{8}}}{{a^{8}-b^{8}}}+
\frac{{a^{8}-b^{8}}}{{a^{8}+b^{8}}}$

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So,, I got this problem and the first thing I did was factoring
and I found the common denominator of 't'

Then it turned out that

$t=\frac{{2(a^{4}+b^{4})}}{{a^{4}-b^{4}}}$

Also I could change 'the follwing' into this.

$\frac{{2(a^{16}+b^{16})}}{{a^{16}-b^{16}}}$

Next, I tried dividing 'the following' by 't' so that I can find out what I should multiply to 't' to make it as 'the following'.

$\frac{{(a^{16}+b^{16})}({a^{4}-b^{4})}}{({a^{16}-b^{16})}({a^{4}+b^{4})}}$

.... And now I'm stuck.
I felt like I was getting closer to the answer but....

Any help would be appreciated..

2. We have $t=\frac{2(a^4+b^4)}{a^4-b^4}$

$a^4t-b^4t=2a^4+2b^4\Rightarrow\left(\frac{a}{b}\right)^ 4=\frac{t-2}{t+2}\Rightarrow\left(\frac{a}{b}\right)^8=\left (\frac{t-2}{t+2}\right)^2$.

Now, $\frac{a^8+b^8}{a^8-b^8}+\frac{a^8-b^8}{a^8+b^8}=\frac{\left(\frac{a}{b}\right)^8+1}{ \left(\frac{a}{b}\right)^8-1}+\frac{\left(\frac{a}{b}\right)^8-1}{\left(\frac{a}{b}\right)^8+1}$

Replace $\left(\frac{a}{b}\right)^8$ with $\left(\frac{t-2}{t+2}\right)^2$