Find all $x\in\mathbb{R}$ such that $\left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x$.

Solution 1:

It is not difficult (formula for the double angle) to show that $$\sin \left(\frac{ \pi }{8} \right)= \sqrt{ \frac{2- \sqrt{2} }{4} }$$ which in combination with the trigonometric one gives $$\cos^2\left(\frac{ \pi }{8} \right)=1-\sin^2\left(\frac{ \pi }{8} \right)=\frac{2+ \sqrt{2} }{4}\Rightarrow \cos \left( \frac{ \pi }{8} \right)= \sqrt{ \frac{2+ \sqrt{2} }{4} }$$ thus our equation can be expressed equivalently in the form $$\left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x$$ $$\left( \sqrt{\frac{2- \sqrt{2} }{4}}\right)^x+\left( \sqrt{\frac{2+ \sqrt{2} }{4}}\right)^x=1 $$ $$\sin^x\left( \frac{ \pi }{8} \right)+\cos^x \left( \frac{ \pi }{8} \right)=1$$ of course thanks to the trigonometric one $x=2$ is a trivial solution. Uniqueness of this solution is due to the fact $\sin \& \cos \le 1$. Formally, you can consider cases $x>2$ or $x<2$ and estimate the left side.

Solution 2:

Another way.

Rewrite our equation in the following form: $$\left(\sqrt{\frac{2-\sqrt2}{2+\sqrt2}}\right)^x+1=\left(\frac{2}{\sqrt{2+\sqrt{2}}}\right)^x.$$ We see the the left side decreases and the right side increases,

which says that our equation has one real root maximum.

But $2$ is a root and we are done!

Solution 3:

Divide your original equation by $2^x$ getting

$((\sqrt{2+\sqrt2}/2)^x+((\sqrt{2-\sqrt2}/2)^x=1.$

As $\sqrt2<2$, both terms on the left side are exponential functions with base strictly between $0$ and $1$. Therefore, the left side is strictly decreasing forcing no more than one solution. Thus any solution that might be found by inspection must be the only one.