How do I prove that group determined by generators and relations is trivial?

Question

I have a group $\langle a,b\mid a^4=b^2=1, ab^2=b^3a, ba^3=a^2b\rangle $ how do I show that it's trivial?

Answer 1

$a=ab^2=b^3a \implies b^3=1$. Then $b^2=1$ and $b^3=1$ imply that $b=1$. Then $ba^3=a^2b \implies a^3=a^2 \implies a=1$.

Answer 2

Since $b^2=1$,$$ab^2=b^3a\implies a=ba\implies b=1$$Then using this, $$ba^3=a^2b\implies a^3=a^2\implies a=1$$