Question

calcule
a) cos 105. b) tg 15

Answer 1

cos 105° =
cos (60 + 45) =
cos60.cos45 - sen60.sen45 =
1/2 . √2 /2 - √3 /2 . √2 /2 =
√2 /4 - √6 / 4 ]=
(√2 - √6) / 4 

tg 15 = tg (45 - 30)

$tg 15 = \frac{tg45 - tg 30}{1 + tg45.tg30} = \frac{1 - \frac{ \sqrt{3} }{3} }{1 + 1 . \frac{ \sqrt{3} }{3} } = \frac{ \frac{3- \sqrt{3} }{3} }{ \frac{3 + \sqrt{3} }{3} } = \frac{3- \sqrt{3} }{3} . \frac{3}{3 + \sqrt{3} } = \frac{3 - \sqrt{3} }{3 + \sqrt{3} } . \frac{3 - \sqrt{3} }{3 - \sqrt{3} } = \\ \frac{9 - 6 \sqrt{3}+3 }{9 - 3} = \frac{12 - 6 \sqrt{3} }{6} = \frac{6 . (2 - \sqrt{3} )}{6} = 2 - \sqrt{3}$

Logo tg 15 = 2 - √3