Question
Sat November 19, 2011 By:
 

prove cos^2 5+ cos^2 10 + cos^2 15....cos^2 90 = 17/2

Expert Reply
Sat November 19, 2011
cos25+cos210+cos215......cos290=17/2
 
Starting from LHS
 
LHS = cos25+cos210+cos215......cos290
 
= cos25+cos210+cos215......cos275+cos280+cos285+cos290
 
Now cos90= 0, putting the value we get
 
= cos25+cos210+cos215......cos275+cos280+cos285
 
= (cos25+cos285)+(cos210+cos280)+(cos215+cos285)+....+(cos240+cos250)+cos245
 
Now put cos?=sin(90-?), replacing this we get
 
= (cos25+sin25)+(cos210+sin210)+(cos215+sin215)+....+(cos240+sin240)+cos245
 
Total number of terms in the equation initially were 90/5 = 18
cos 90 was zero so remaining terms were 17, cos 45 is separate so remaining terms 
are 16 which are grouped in group of 2 so means 8 groups.
 
since cos2x+sin2x=1 we have
 
= (1)+(1)+(1)....8 times + cos245
 
now cos45=1/?2 putting the value we get
 
= 8 + 1/2 = 17/2 = RHS 
 
Proved
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