Find all $x$ in in the interval $[0, 2π]$ that satisfy the equation:
$$4\cos^2(x) - \sin^2(2x) + 5\sin^2(x) = 4.$$
I found the solution with a graphing calculator but I need the way of finding it without one. I have tried everything I could think of in the past 2 days but I'm still stuck.
I have tried things like:\begin{align*} \sin^2(x) + \cos^2(x) = 1\\ \sin(2x) = 2 \times \sin(x)\times \cos(x) \end{align*} which in my place will be:$$ \sin^2(2x) = 4 \times \sin^2(x) \times \cos^2(x). $$
But after using one of them I get stuck a bit later.
The answer using graphing calculator is: $0, π/3, 2π/3, π, 4π/3, 5π/3, 2π$.
Use the trig identity to replace $\sin(2x)$ with $2\sin(x)\cos(x)$. You will now have an equation with $\sin^2(x)$ and $\cos^2(x)$.
Now use the Pythagorean identity $\sin^2(x)+\cos^2(x)=1$ to get an equation with only $\sin^2(x)$ or only with $\cos^2(x)$. It will be a quadratic equation in that expression. (Hint: the quadratic equation will be a little easier to solve if you convert everything to $\sin^2(x)$.)
Solve that quadratic equation, then find $\sin(x)$ or $\cos(x)$ (depending on what you chose in the previous step), then find $x$.
So you had the right identities but apparently did not see just where to use them. You should be able to finish from here.