<s>[INST] Apporoaches to solve the given algebraic expression If  $\displaystyle \ \  x^{4} \ +\ x^{2} \ =\ \frac{11}{5}$  then what is the value of the given expression
$$\displaystyle \left(\frac{x+1}{x-1}\right)^{\frac{1}{3}} +\ \left(\frac{x-1}{x+1}\right)^{\frac{1}{3}} \ =\ \ ?$$
My Try :
As I can find the value of $\displaystyle x$, from the given equation but it will be tedious I think !.
$$\displaystyle x^{4} \ +\ x^{2} \ =\ \frac{11}{5}$$
$$\displaystyle \Longrightarrow \ x^{2} +1/2 \ =\ \frac{7}{2\sqrt{5}}$$
$$\displaystyle \Longrightarrow \ x^2 \ =\  \ \frac{7-\sqrt{5}}{2\sqrt{5}}$$
Which is getting too much complicated to solve the expression by putting the value of $\displaystyle x$.
What could be the other way to solve the given expression?
 [/INST] Let me try it
we have putting $t=\tan\Big(\frac{x_1}{2}\Big)$
$$\cos(x_1)=\frac{1-t^2}{1+t^2}$$
It's the Weierstrass substitution
Putting $y=t^2$ we get :
$$\cos(x_1)=\frac{1-y}{1+y}$$
Now we put $x=y$ to get :
$$\Big(-\cos(x_1)\Big)^{\frac{1}{3}}+\Big(-\frac{1}{\cos(x_1)}\Big)^{\frac{1}{3}}=?$$
After that I have tried $\cos(3x)=4\cos^3(x)-3\cos(x)$.
Hope it helps (I think it's hard).
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