Relationship between elliptic integrals of first and second order. A colleague and I are making an issue where order to reach the result we are asked to prove identity between elliptic integrals first and second order.

This is:

$$\frac{V}{\pi}\left( \frac{\pi}{2}-\sqrt{1-k^2}K(k) \right)=\frac{V}{\pi}\left(E(k) -(1-k^2)K(k) \right)$$

Where $E (k)$ and $K (k)$ is the elliptic integral of second and first order, respectively.

In addition: $V$ is constant. Our question is: Is it true identity? What if so, as demonstrated? Any help is the grateful.
It's not an identity. Try a single value, $k\neq0$ and see.
It's not an identity. Try a single value, $k\neq0$ and see.