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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.

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

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

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

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

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

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\[ \large \sin(\alpha) + \sin(3\alpha) + \sin(4\alpha) + \sin(5\alpha) + \sin(9\alpha) \]

Let \( \alpha = \dfrac{2\pi}{11} \), find the closed form of the expression above.

Give your answer to 3 decimal places.

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