Miscellaneous
Last updated at April 16, 2024 by Teachoo
Misc 3 Prove that: (cos π₯ + cos y)2 + (sin π₯ β sin y)2 = 4cos2 (π₯ + y)/2 Solving LHS (ππ¨π¬β‘π+ππ¨π¬β‘π )^π + (π¬π’π§β‘πβπ¬π’π§β‘π )^π = cos2 π₯+cos2 π¦+2 cosβ‘π₯ cosβ‘π¦+sin2 π₯+ sin2 π¦ β 2sinβ‘π₯ sinβ‘π¦ = (ππππ π+ππππ π)+(ππππ π+ππππ π )+2 (cosβ‘π₯ cosβ‘π¦ βsinβ‘π₯ sinβ‘π¦) = 1 + 1 +2 (ππ¨π¬β‘π ππ¨π¬β‘πβπ¬π’π§β‘π π¬π’π§β‘π ) = 2 + 2 πππβ‘(π + π) = 2 [1 + πππβ‘(π + π) ] = 2 [1+2cos2 ((π₯ + y)/2) β 1 ] = 4 cos^2 ((π₯ + y)/2) = RH.S. Hence proved