Solve this equation. \(\displaystyle{\left(\sqrt{{\sqrt{{{2}}}+{1}}}\right)}^{{{\sin{{\left({x}\right)}}}}}+{\left(\sqrt{{\sqrt{{{2}}}-{1}}}\right)}^{{{\sin{{\left({x}\right)}}}}}={2}\)



Answered question


Solve this equation.

Answer & Explanation

Ashley Olson

Ashley Olson

Beginner2022-03-24Added 12 answers

To solve
let us rewrite as
First consider the auxiliary equation
You can verify this has only one solution, y=1. Next, notice
By laws of exponents, our original equation is then
which we recognize as the auxiliary equation with y=(2+1)12sin(x). Therefore the solution to our original equation has
Since 2+1>1, we must have
The solutions are the integer multiples of π

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