Clausen's formula

Remarks

One proof compares differential equations. If $y$ solves a second order linear equation, then $y^2$ solves a third order linear equation, the symmetric square. For the hypergeometric equation with $c = a + b + \frac12$, the symmetric square turns out to be the generalized hypergeometric equation of the ${}_3F_2$ on the right. Both sides are analytic at $0$ with value $1$, and that equation has only one such solution, so they agree. Check of the first coefficient: on the left, the coefficient of $x$ is $\frac{2ab}{a + b + \frac12}$. On the right it is $\frac{2a \cdot 2b \cdot (a + b)}{(a + b + \frac12)(2a + 2b)} = \frac{2ab}{a + b + \frac12}$, as it should be. For $a = b = \frac14$ the formula reads ${}_2F_1(\frac14, \frac14; 1; x)^2 = {}_3F_2(\frac12, \frac12, \frac12; 1, 1; x)$. The relation $c = a + b + \frac12$ between the parameters is what makes the symmetric square equation hypergeometric, and the proof uses it.

Going deeper

For a student

The formula links two levels of the hypergeometric hierarchy, ${}_2F_1$ and ${}_3F_2$. It gives positivity results: when the ${}_2F_1$ is real, the ${}_3F_2$ on the right is a square, hence nonnegative. It is also used to turn identities for squares of elliptic integrals into series for $1/\pi$.