1. ## Trigonomertric identities

[tex]cos20°cos100°+cos100°cos140°-cos140°cos200°=\frac{-3}{4}[tex]

2. Originally Posted by matsci0000
[tex]cos20°cos100°+cos100°cos140°-cos140°cos200°=\frac{-3}{4}[tex]
$cos20°cos100°+cos100°cos140°-cos140°cos200°$

= $cos20°cos100°+cos100°cos140°+cos140°cos20°$

Now, see $
sec\frac{\pi}{9},sec\frac{5\pi}{9},sec\frac{7\pi}{ 9}
$
are the roots of equation

$
sec^{3}\theta+6sec^{2}\theta-8=0
$

hence, $sec\frac{\pi}{9}*sec\frac{5\pi}{9}*sec\frac{7\pi}{ 9}=8$

= $cos20°cos100°+cos100°cos140°+cos140°cos20°$

= $sec\frac{\pi}{9}*sec\frac{5\pi}{9}*sec\frac{7\pi}{ 9} \left(
sec\frac{\pi}{9}+sec\frac{5\pi}{9}+sec\frac{7\pi}{ 9} \right)
$

= $\frac{-6}{8}$

= $\frac{-3}{4}$