Demonstrate equality (cotangent)

Question

Any hint to demonstrate that the function

$$ C(z) = \pi\cot \pi z $$

satisfies

$$ 2F(2z) = F(z) + F(z + \frac{1}{2}), $$ please?

Thanks.

Answer 1

$$ C(z) +C(z+{1/2})= \pi\Big(\cot (\pi z)+\cot( {\pi z+{\pi/2}})\Big) =\pi\Big(\cot (\pi z)-\tan( \pi z)\Big) $$ $$ = \pi{\cos^2 (\pi z)-\sin^2( \pi z)\over \cos (\pi z)\sin( \pi z)} =\pi {2\cos(2\pi z)\over \sin(2\pi z)} = 2C(2z) $$