Geometry

Proving Trigonometric Identities

Proving Trigonometric Identities: Level 4 Challenges

         

tan21+tan23+tan25++tan287+tan289= ?\large \tan^2 1^\circ + \tan^2 3^\circ + \tan^2 5^\circ + \cdots+ \tan^2 87^\circ + \tan^2 89^\circ = \ ?

sec(π10)sec(3π10)sec(7π10)sec(9π10)=  ?\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 ) = \; ?

12x=cos(a)cos(2a)cos(3a)cos(999a)\large \dfrac1{2^x} = \cos (a) \cos(2a) \cos(3a) \cdots \cos(999a)

The equation above holds true for a=2π1999a = \dfrac{2\pi}{1999} . Find xx.

tanπ7tan2π7tan3π7=A\large \tan\frac{\pi}{7}\tan\frac{2\pi}{7}\tan\frac{3\pi}{7}= \sqrt{A} Find AA.

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

If f(2π1999)=12kf \left(\dfrac{2\pi }{1999}\right) = \dfrac{1}{2^{k}}, find kk.

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