Trig Power Challenge

Geometry Level 3

If the value of sin24π24+cos24π24\sin^{24}\frac{\pi}{24} + \cos^{24}\frac{\pi}{24}

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

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

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