Let x = $cos\theta + isin\theta $                                                        

$\therefore\dfrac{1}{x} = cos\theta - isin\theta$

Thus,$ x^{n} = cos\theta + isin\theta$

$\dfrac{1}{x^ {n}} = cos\theta - isin\theta$

$\therefore x + \dfrac{1}{x} = 2cos\theta$

$x - \dfrac{1}{x} = 2isin\theta$ and

$x^{n} + \dfrac{1}{x^ {n}} = 2cosn\theta,$

$x^{n} - \dfrac{1}{x^ {n}} = 2isinn\theta$

NOW,

$(2cos\theta)^ {6} = (x + \dfrac{1}{x} )^{6}$=$ x^{6} + 6x^{5}\times\dfrac{1}{x} + 15x^ {4}\times\dfrac {1} {x^ {2}} + 20x^{3}\times\dfrac {1} {3^ {3}} + 15x^ {2}\times\dfrac {1} {x {4}} + 6x\times\dfrac {1}{x^ {5}} + \dfrac {1}{x^ {6}} 2^ {6}cos^{6}\theta = x^{6} + 6x^{4} + 15x^ {2} + 20 + 15x^ {2} + \dfrac {6}{x^ {4}} + \dfrac {1}{x^ {6}}$…………………………………………….(1)

$(2isin\theta)^{6} = (x - \dfrac{1}{x} )^{6}$= $x^{6} - 6x^{5}\times\dfrac{1}{x} + 15x^ {4}\times\dfrac {1} {x^ {2}} - 20x^{3}\times\dfrac {1} {3^ {3}} + 15x^ {2}\times\dfrac {1} {x {4}} - 6x\times\dfrac {1}{x^ {5}} + \dfrac {1}{x^ {6}} -2^ {6}sin^{6}\theta = x^{6} - 6x^{4} + 15x^ {2} - 20 + \dfrac{15}{x^ {2}} - \dfrac {6}{x^ {4}} + \dfrac {1}{x^ {6}}$ …………………………………………….(2)

Adding equation (1) and (2),we get

$2^ {6}cos^{6}\theta -2^ {6}sin^{6}\theta = 2x^{6} + 30x^ {2} + 30x^ {2} - \dfrac {30}{x^ {2}} + \dfrac {2}{x^ {6}}$

$2^{6}(cos^{6}\theta - sin^{6}\theta) = 2(x^{6} + \dfrac {1}{x^ {6}} ) + 30(x^{2} + \dfrac {1}{x^ {2}} ) $

$(cos^{6}\theta - sin^{6}\theta) = \dfrac {1}{2^ {5}}((2cos\theta)^ {6}) + \dfrac {30}{2^ {6}}(2cos2\theta)$

$\therefore(cos^{6}\theta - sin^{6}\theta) = \dfrac {1}{16}(cos6\theta + 15cos2\theta)$