# Question 6f9af

Jan 29, 2017

1)
$\cos {75}^{\circ}$
$= \cos \left(45 + 30\right)$

$= \cos 45 \cos 30 - \sin 45 \sin 30$

$= \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \times \frac{1}{2}$
$= \frac{\sqrt{3} - 1}{2 \sqrt{2}}$

2)
$\sin {75}^{\circ}$
$= \sin \left(45 + 30\right)$

$= \sin 45 \cos 30 + \cos 45 \sin 30$

$= \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \times \frac{1}{2}$
$= \frac{\sqrt{3} + 1}{2 \sqrt{2}}$

3)
$\sin \left(\frac{17 \pi}{6}\right) \cos \left(\frac{8 \pi}{3}\right) - \cos \left(\frac{17 \pi}{6}\right) \sin \left(\frac{8 \pi}{3}\right)$

$= \sin \left(\frac{17 \pi}{6} - \frac{8 \pi}{3}\right)$

$= \sin \left(\frac{17 \pi}{6} - \frac{16 \pi}{6}\right)$

$= \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$

4)

cos((7pi)/5))cos((3pi)/5)-sin((7pi)/5))sin((3pi)/5)#

$= \cos \left(\frac{7 \pi + 3 \pi}{5}\right)$

$= \cos \left(2 \pi\right) = \cos 0 = 1$

5)

$\frac{\tan 56 - \tan 26}{1 + \tan 56 \tan 26}$
$= \tan \left(56 - 26\right) = \tan 30 = \frac{1}{\sqrt{3}}$

6)

$\frac{\tan \frac{\pi}{12} + \tan \frac{\pi}{6}}{1 - \tan \frac{\pi}{12} \tan \frac{\pi}{6}}$

$= \tan \left(\frac{\pi}{12} + \frac{\pi}{6}\right)$

$= \tan \left(\frac{\pi + 2 \pi}{12}\right)$

$= \tan \left(\frac{\pi}{4}\right) = 1$