# Integral of Legendre polynomials with tangent

I have encountered the following relationship$$^{[1][2]}$$, stated without proof both times

$$\int_0^\gamma dt \tan(t/2)\cdot [P_n(\cos(t))+P_{n-1}(\cos(t))]=\frac{1}{n}[P_{n-1}(\cos(\gamma))-P_{n}(\cos(\gamma))]$$

Where $$P_n(x)$$ is the $$n^{th}$$ Legendre polynomial, and $$\gamma<\pi$$.

I've tried integration by parts, substitution of $$u=\cos(t)$$, and looked in Gradshteyn. Mathematica doesn't integrate it (at least, not in the current form). I've verified that the relationship is correct for modest values of $$n$$. I suspect that there is some property of the Legendre polynomials that facilitates this, but I don't see what it is.

[1] Mixed boundary value problems, Dean G. Duffy, eq. 3.1.80, p95

Since $$\tan(x/2)=\frac{\sin(x)}{1+\cos(x)}$$, by letting $$\cos(\gamma)=T\in[-1,1]$$ your identity can be written in the simplified form
$$\int_{T}^{1}\frac{dx}{1+x}(P_n(x)+P_{n-1}(x))\,dx = \frac{1}{n}(P_{n-1}(T)-P_n(T)).$$
The identity clearly holds at $$T=1$$. By differentiating both sides wrt $$T$$ we find Bonnet's identity and we are done.