\(sin^8 \left(\frac{5\pi}{12}\right) - cos^8 \left(\frac {5\pi}{12}\right) = ?\)
\(\frac{5\pi}{12} = \frac{5 \times 180°}{12} = 5\times 15° = 75° \)
\(sin\left(\frac{5\pi}{12}\right) = sin75° + sin(30° + 45°)\)
\(= sin30° cos45° + cos30° sin45°\)
\( = \frac12 \times \frac 1{\sqrt2} + \frac{\sqrt 3}2 \times \frac 1{\sqrt 2}\)
\(= \frac 1{2\sqrt 2} ( 1 + \sqrt 3)\)
\(cos\left(\frac {5\pi}{12}\right) = cos 75° = cos(30° + 45°)\)
\(= cos 30° cos 450° - sin30° sin45°\)
\(= \frac {\sqrt 3}2 \times \frac 1{\sqrt 2} - \frac 12 \times \frac1{\sqrt 2}\)
\( = \frac{\sqrt 3 - 1}{2\sqrt 2}\)
\(\therefore sin^8\left(\frac {5\pi}{12}\right) - cos^8 \left(\frac {5\pi}{12}\right) = \left(\frac{\sqrt 3 + 1}{2\sqrt 2}\right)^8 - \left(\frac{\sqrt 3 - 1}{2\sqrt 2}\right)^8\)
\(= \left(\frac 1{2\sqrt 2}\right)^8 \left((\sqrt 3 + 1)^8 - (\sqrt 3 - 1)^8\right)\)
\(= \frac 1{2^8 .2^4} \left(2(8.(\sqrt 3)^7 +56(\sqrt3)^5 +56(\sqrt 3)^3 + 8\sqrt 3)\right)\)
\(= \frac 1{2^{12}} . 16 ((\sqrt 3)^7 + 7(\sqrt 3)^5 + 7(\sqrt 3)^3 + \sqrt 3)\)
\(= \frac {\sqrt 3}{256} ((\sqrt 3)^6 + 7(\sqrt 3)^4 + 7(\sqrt 3)^2 + 1)\)
\(= \frac{\sqrt 3}{256}(27 + 63 + 21 + 1)\)
\(= \frac{\sqrt 3 \times 112}{256}\)
\(= \frac{7\sqrt 3}{16}\)
