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Trignometry

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19 Jan 2008 19:06:05 IST

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Prove that 1/sin2x + 1/sin4x + ...+1/sin2ⁿx = cotx - cot2ⁿx

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Rohit Gupta

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Joined: 10 Jan 2008

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19 Jan 2008 23:01:38 IST

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cot x - cot 2x = cos x/sin x - cos 2x/sin 2x
                    = (2(cos x)^2 - cos 2x) / sin 2x
                    = (2(cos x)^2 - (2(cos x)^2 - 1)) / sin 2x
                    = 1/sin 2x

similarly (cot 2x - cot 4x) = 1/sin 4x
             (cot 4x - cot 8x) = 1/sin 8x
             ......................................
             (cot (2^n-1)x - cot (2^n)x) = 1/sin (2^n)x

Now,

in question, R.H.S. = cot x - cot (2^n)x

                          = (cot x - cot 2x) + (cot 2x - cot 4x) + (cot 4x - cot 8x) + .....+ (cot (2^n-1)x - cot (2^n)x)

                            = 1/sin 2x + 1/sin 4x + ............ + 1/sin (2^n)x

                            = L.H.S. Done!

Hari Shankar

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Posts: 2173

20 Jan 2008 07:23:49 IST

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Sprinkle Sir, good soln. But cud we wait for the aspiring students to have a go at it before we post our solutions.

Brad

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Joined: 9 Sep 2007

Posts: 647

20 Jan 2008 08:02:11 IST

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Ya i agree with hsbhatt, Sprinkle sir our Big Brother over here.......is already through so wouldn't it be better that we try and if we fail then Big Brother can always help us
out.I solved it yesterday night same way but because of boards could not post it.I mean there is no doubt about his brilliance but maybe we should be given a little more opportunity.This is totally personal a view let others speak too.

kapeesh mehra

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Joined: 15 Jan 2008

Posts: 8

20 Jan 2008 09:05:55 IST

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i also agree to Greatdreams we all accept that u r a gem and u should post a solun when others can not find it or can not find a good solun

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