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Proving Trigonometric Identities

So, you've memorized your fundamental identities, but how can you prove the more obscure ones? See how to apply the basic building blocks of trig to understand deeper relationships.

Level 4

         

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

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

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

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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