Question

Maths lovers ... i want to test you.. solve it fast​

Answer 1

$answer\: \: \: \\ let \: \: \: theta \: \: = \alpha \\ \\ given \: \: \: \: \cos( \alpha ) = \frac{8}{17} \\ \\ \cos( \frac{\pi } {6 } + \alpha ) + \cos( \frac{\pi}{4} - \alpha ) + \cos( \frac{2\pi}{3} - \alpha ) = \beta \: \: let \\ \\ \beta = \cos( \alpha ) ( \cos( \frac{\pi}{4} ) + \cos( \frac{\pi}{6} ) + \cos( \frac{2\pi}{3} ) ) + \sin( \alpha ) ( \sin( \frac{2\pi}{3} ) + \sin( \frac{\pi}{4} ) - \sin( \frac{\pi}{6} ) ) \\ \\ \beta = \cos( \alpha ) ( \frac{1}{ \sqrt{2} } + \frac{ \sqrt{3} }{2} - \frac{1}{2} ) + \sin( \alpha )( \frac{ \sqrt{3} }{2} + \frac{1}{ \sqrt{2} } - \frac{1}{2} ) \\ \\ \beta = \frac{( \sqrt{3} - 1)}{2} + \frac{1}{ \sqrt{2} } ( \cos( \alpha ) + \sin( \alpha ) ) \\ \\ \beta = \frac{( \sqrt{3} - 1)}{ \sqrt{2} } + \frac{1}{ \sqrt{2} } ( \frac{8}{17} + \frac{15}{17} ) \\ \\ \beta = \frac{( \sqrt{3} - 1)}{2} + \frac{1}{ \sqrt{2} } ( \frac{ 23}{17)} \\ \\ \\ note \: \: \: \: \: \: \: \\ \cos( \alpha + \beta ) = \cos( \alpha ) \cos( \beta ) - \sin( \alpha ) \sin( \beta ) \\ \cos( \alpha - \beta ) = \cos( \alpha ) \cos( \beta ) + \sin( \alpha ) \sin( \beta ) \\and \\ \cos( \frac{2\pi}{3} ) = \frac{ - 1}{2}$