Geometry
# Proving Trigonometric Identities

\[\large \dfrac1{2^x} = \cos (a) \cos(2a) \cos(3a) \cdots \cos(999a) \]

The equation above holds true for \(a = \dfrac{2\pi}{1999} \). Find \(x\).

\[\large \tan\frac{\pi}{7}\tan\frac{2\pi}{7}\tan\frac{3\pi}{7}= \sqrt{A} \] Find \(A\).

\[\large f(x) = \cos(x) \cdot \cos(2x) \cdot \cos(3x)\cdots \cos(999x)\]

If \(f \left(\dfrac{2\pi }{1999}\right) = \dfrac{1}{2^{k}}\), find \(k\).