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Geometry

Proving Trigonometric Identities

Proving Trigonometric Identities: Level 4 Challenges

         

\[\large \tan^2 1^\circ + \tan^2 3^\circ + \tan^2 5^\circ + \cdots+ \tan^2 87^\circ + \tan^2 89^\circ = \ ? \]

\[\large \sec \left ( \dfrac {\pi}{10} \right ) \sec \left ( \dfrac {3\pi}{10} \right ) \sec \left ( \dfrac {7\pi}{10} \right ) \sec \left ( \dfrac {9\pi}{10} \right ) = \; ?\]

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

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 \(k\).

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