Question

X=cos55°,y=cos65°,z=cos175°, find xy+yz+xz

Answer 1

Given x = cos 55, y = cos 65, z = cos 175.

We need to find xy, yz, zx.

= > cos 55 * cos 65 + cos 65 * cos 175 + cos 175 * cos 55

We know that 2cosAcosB = cos(A + B) + cos(A-B).

= > (1/2)(cos(55 + 65) + cos(65 - 55)) + (1/2)(cos(65 + 175) + cos(175 - 65)) + (1/2)(cos(175 + 55)cos(175 - 55))


= > (1/2)(cos120+cos10) + (1/2)(cos240+cos110) + (1/2)(cos230 + cos120)


= > (1/2)(cos120) + (1/2)cos(10) + (1/2)cos240 + (1/2)cos(110) + (1/2)cos(230) + (1/2)cos120


= > (1/2)cos(180 - 60) + (1/2)cos(10) + (1/2)cos(180 + 60) + (1/2)cos(110) + (1/2)(cos230) - 1/2(cos(180 - 60))


= > (1/2)(-cos60) + (1/2)cos(10) + (1/2)cos(-cos60) + (1/2)cos110 +  (1/2)cos230 - 1/2(-cos60)


= > (1/2)(-1/2) + (1/2)cos(10) + (1/2)(-1/2) + (1/2)cos(110) + (1/2)cos(230) - (1/2)(-1/2)


= > -1/4 + 1/2cos(10) - 1/4 + (1/2)cos(110) + (1/2)cos230 - 1/4


= > -(3/4) + (1/2)(cos10 + cos110 + cos230)


$= \ \textgreater \ - \frac{3}{4} + ( \frac{1}{2})( \frac{2cos(110 + 10)}{2} \frac{2cos(110 - 10)}{2} + cos230)$

$= \ \textgreater \ - \frac{3}{4} + \frac{1}{2}(2cos60cos50 + cos(180 + 50))$

$= \ \textgreater \ - \frac{3}{4} + \frac{1}{2} (2cos60cos50 - cos50)$

$= \ \textgreater \ - \frac{3}{4} + \frac{1}{2}cos50(2 * \frac{1}{2} - 1)$

$= \ \textgreater \ - \frac{3}{4} + \frac{1}{2} cos50(0)$

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



Hope this helps!

Answer 2

Answer:

-3/4

Step-by-step explanation:

Hpe it helps