Geometry
# Trigonometric Functions

\[ \displaystyle \sum_{k=1}^{145} \left [ \sin \left ( \frac {2\pi k}{8} \right ) - i \ \cos \left ( \frac {2\pi k}{14} \right ) \right ] = \frac {1}{\sqrt{a}} - i \cos \left ( \frac {\pi}{b} \right ).\] Find the sum of the positive integers \(a\) and \(b.\)

**Note:** \(i = \sqrt{-1}.\)

Evaluate \[\large \sum_{m=1}^{\infty} \tan^{-1} \left(\frac{8m}{16m^4-32m^2+5}\right).\]

Remember that the range of the inverse tangent function is \( ( - \frac{ \pi}{2} , \frac{ \pi }{2}) \).

If the value of \[\sin^{24}\frac{\pi}{24} + \cos^{24}\frac{\pi}{24}\]

is expressed in simplest form as \( \dfrac{a + b\sqrt{c}}{d},\) find the last three digits of \(a + b + c + d.\)

**Note:** Simplest form refers to the condition that \(a,b,c,d\) are positive integers with \(\gcd(a,b,d)=1\) and \(c\) is not divisible by the square of any prime.

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