 Solution

Put $R(z)=\frac{1}{4}({\wp }^{\prime \prime }/{\wp }^{\prime }{)}^2-2\wp (z)$. This is an elliptic function. Its only possible poles modulo $\Lambda$ are $0$ and the three simple zeros $h$ of ${\wp }^{\prime }$. At such a nonzero half-period, periodicity and evenness give $\wp (h+t)=\wp (h-t)$, soIt follows that $R(h+t)=1/(4t^2)+O(1)$ with no simple-pole term. This is exactly the principal part of $\wp (2h+2t)=\wp (2t)$.

At zero the source expansion yieldsConsequently $R(z)=1/(4z^2)+O(z^2)$, again matching $\wp (2z)$. Thus $R(z)-\wp (2z)$ is a holomorphic function on a compact connected Riemann surface, hence constant. Its limit at zero is zero, provingThe equality extends over the apparent singularities as an identity of meromorphic functions.