Question

cos24 + cos48 - cos84 - cos12

Answer 1

cos(24)+cos(48) = 2cos(36)cos(12)
cos(84)+cos(12) = 2cos(48)cos(36)
cos(24)+cos(48)-(cos(84)+cos(12) ) = 2cos(36)(cos(12)-cos(48)) = -4cos(36)sin(30)sin(-18) = 2cos(36)sin(18) = 2cos(36)sin(36)sin(18)/sin(36) = sin(72)sin(18)/sin(36) = sin(72)cos(72)/sin(36) = (1/2)*sin(144)/sin(36) = 1/2

Answer 2

cos24 + cos48 - cos84 - cos12

Using the trigonometric identity cos(A+B) = cosA*cosB - sinA*sinB, we can rewrite the equation as:

cos24 + cos48 - cos84 - cos12 = cos(60-36) + cos(60+12) - cos(90+6) - cos(60-48) = cos60*cos36 - sin60*sin36 + cos60*cos12 - sin60*sin12 - cos90*cos6 + sin90*sin6 - cos60*cos48 + sin60*sin48 = (1/2)*(√5/2) - (√3/2)*(1/2) + (1/2)*(√3/2) - (√3/2)*(1/2) - 0 - 1 + (1/2)*(1/2) - (√3/2)*(√3/2) = (√5/4) - (√3/4) + (√3/4) - (√3/4) - 1/4 - 1/2 + 1/4 - 3/4 = -1/2

So, the value of cos24 + cos48 - cos84 - cos12 is -1/2.