Cool equation

Algebra Level 3

How many ordered solutions $(x ,y)$ are there for the equation:

${2}^{{x}^{2}} + {2}^{{y}^{2}} = 1 + {cos}^{2} x$

1 2 4 3

1 solution

Adrian Neacșu
Apr 25, 2014

We know that ${2}^{a} \ge 1$ if $a\ge 0$ .

Then ${2}^{{x}^{2}} + {2}^{{y}^{2}} \ge 2$ since ${x}^{2} \ge 0$ and ${y}^{2} \ge 0$ .

But $1 + {cos}^{2} x \le 2$ .

Then for the equation be need both LHS and RHS to be equal to 2.

Which gives $x = y = 0$