Find side length $a$ of triangle given $\cos(2α − β) + \sin(α + β) = 2$ and $b = 2\sqrt3$

Question

Triangle with angles $\alpha \beta \gamma$ and length sides $a b c$ across each angles.
cos(2α − β) + sin(α + β) = 2 and b = 2√3

What is a ?

I draw it

$b^2 = a^2 + c^2 -2ac\cos\beta$

cos(2α − β) = cos2asinb + sin2asinb

What is the simple way to solve it?

Answer 1

$\cos(2α − β) + \sin(α + β) = 2$ Is true only if $\cos(2α − β)=1$ and $\sin(α + β) =1$. Then,

$$ 2α − β = 0,\>\>\>\>\> α + β=90$$

which yields $α =30$ and $β =60$, i.e. a right triangle. Thus, given $b=2\sqrt3$, we obtain $a = b \tan 30= 2$.