Question

solve the equation cos(2x)= √2-cos(2x)

Answer 1

square both sides, plugin cos(2x) = u, solve the quadratic

Answer 2

You add cos(2x) to both sides:

\[\cos(2x)+\cos(2x) = \sqrt{2}\]
\[2\cos(2x) = \sqrt{2}\]
\[\cos(2x) = \frac{\sqrt{2}}{2}\]
\[\cos(2x)=cos( \frac{π}{4})\]
\[2x=2kπ+\frac{π}{4} (Equation\ 1)\]
or \[2x=2kπ-\frac{π}{4}(Equation\ 2)\]

And you solve both for x.

Answer 3

after pluggin in cos2x=u,u get a quadratic equation in u, u^2+u-2=0,solving it,u get u=2,-1.therefore cos2x=2,-1.but cos2x,with its range restricted to [-1,1] cannot be equal to 2.so,cos2x=-1.which gives