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![[Post New]](/templates/default/images/icon_minipost_new.gif) 18 Feb 2007 08:20:32 IST
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Prove that : tan2( /16) + tan2(2/16)+tan2(3/16)+tan2(4/16)+tan2(5/16)+tan2(6/16)+tan2(7/16) = 35 .
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18 Feb 2007 09:44:06 IST
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group two terms at a time and solve <group the terms whose angles add up to pi/2
eg. tan(pi/16)2 + tan(7pi/16)2 = (tan(pi/16)+tan(7pi/16))2-2(tan(pi/16)*tan(7pi/16))
= (tan(pi/16) + tan(7pi/16))2 -2
similarly,
tan(3pi/16)2 + tan(5pi/16)2 = (tan(3pi/16) + tan(5pi/16))2 -2
tan(2pi/16)2 + tan(6pi/16)2 = (tan(2pi/16) + tan(6pi/16))2 -2
Now,
(tan(pi/16) + tan(7pi/16))2 = (sin(7pi/16+pi/16)/(cos(pi/16)cos(7pi/16))2
= 1/((sin(7pi/16)cos(7pi/16))2
= 4/sin2 (pi/8)
similarly,
(tan(2pi/16) + tan(6pi/16))2 =4/sin2(pi/4)
(tan(3pi/16) + tan(5pi/16))2 =4/sin2(6pi/16)
adding everything,
4/sin2 (pi/8) -2 + 4/sin2(pi/4) -2 + 4/sin2(6pi/16) -2 + tan2 pi/4
= 4/sin2 (pi/8) + 4/sin2(6pi/16) +3
= 4( 1/sin2 (pi/8) + 1/sin2 (3pi/8)) +3
= 4( sin2 (pi/8) + cos2 (pi/8)/(sin2 (pi/8)*cos2 (pi/8)) +3
=4(4/4(sin2 (pi/8)*cos2 (pi/8)) +3
= 16/sin2 (pi/4) + 3
= 16*2 +3
= 35
That solves the problem!!
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18 Feb 2007 15:01:57 IST
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I think amaron is right.
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The Scientist does not study nature because it is useful; he studies it because he delights in it, & he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, life would not be worth living. Ofcourse I do not here speak of that beauty that strikes the senses, the beauty of qualities & appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmoniuos order of the parts, & which a pure intelligence can grasp. |
this reply: 2 points
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