There are many books with a proof of Legendre's duplication formula:
$$2^{1-2z}\sqrt{\pi}\Gamma(2z)=\Gamma(z)\Gamma\left(z+\frac{1}{2}\right).$$
which start with the Beta function in the form:
$$B(z_1,z_2)=\int_{0}^{1}u^{z_1-1}(1-u)^{z_2-1}\text{d}u$$
Is there, however, a way to prove it from the trigonometric definition of the Beta function:
$$B(z_1,z_2)=2\int_0^{\frac {\pi}{2}}\sin ^{2z_1-1}\theta\cos^{2z_2-1}\theta\text{d}\theta$$
without going back to the non-trig version of the function?
My attempt so far:
Let $z_1=z_2=z$
$$B(z,z)=2\int_0^{\frac {\pi}{2}}\sin ^{2z-1}\theta\cos^{2z-1}\theta\text{d}\theta$$
$$=2\int_0^{\frac {\pi}{2}}\sin ^{2z_1-1}\theta\cos^{2z_2-1}\theta\text{d}\theta$$
$$=2\int_0^{\frac {\pi}{2}}\left(\frac {\sin 2\theta}{2}\right)^{2z-1}\text{d}\theta$$
$$=2^{-2z}\int_0^{\frac {\pi}{2}}\left(\sin 2\theta\right)^{2z-1}\text{d}\theta$$
and I'm not sure where to go from here.