(1)
cos21°+cos23°+cos35°+cos27°+…+cos285°+cos287°+cos289°In the given expression, we can seethat the angles are 1°, 3°, 5°, 7°,9°, …, 81°, 83°, 85°, 87°, 89°
Let the number of terms in theexpression be n, d be the commondifference, and tn be the nth angle.
∴a+(n–1)d=tn⇒1+(n–1)2=89⇒n = 45$∴cos^2 1°+ cos^2 3°+ cos^3 5°
+ cos^2 7°+ … + cos^2 85°+ cos^2 87°+ cos^2 89°$
=cos21°+cos289°+cos23°+cos287°+cos25°+cos285°+…+cos241°+cos249°+cos243°+cos247°+cos245°$= cos^2 1°+ sin^2 1°+ cos^2 3°+ sin^2 3°
+ … + cos^2 43°+ sin^2 43°+ cos^2 45°$
=[1+1+1+…+1](22times)+[]2= 45/2