Symmetric potential well different solutions

I have solved $H|psirangle=E_{n}|psirangle$ with $V(x)=0$ from $-a<x<a$ and $infty$ otherwise.
If I propose a solution of the form $psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the solution
$$
psi(x)
= frac{1}{sqrt{L}}
sin left(
frac{npi}{2} left(frac{x}{L}-1 right)
right)$$
for $n$ natural.
If I propose a solution of the form
$$psi(x)=A_{n}sin(kx)+B_{n}cos(kx) , .$$
I arrive to the solution
$$
psi(x)
=begin{cases}
frac{1}{sqrt{a}}cos left(frac{npi x}{2a} right) & text{n odd}
frac{1}{sqrt{a}}sin left(frac{npi x}{a} right) & text{n even }
end{cases}
$$

Both solutions with the same energies. However, when plotting both solutions, I see they’re not equal. But I cannot find any mistakes on my procedures.