  Lie bialgebra structures on the extended affine Lie algebra
$\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie
bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be
triangular coboundary. This result is obtained by employing some techniques,
which may also work for more general extended affine Lie algebras, to prove the
triviality of the first cohomology group of $\widetilde{sl_2(\mathbb{C}_q)}$
with coefficients in the tensor product of its adjoint module, namely,
$H^1(\widetilde{sl_2(\mathbb{C}_q)},\widetilde{sl_2(\mathbb{C}_q)}\otimes\widetilde{sl_2(\mathbb{C}_q)})=0$.
