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Trignometry
20 Oct 2008 09:32:23 IST
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22 Oct 2008 21:47:18 IST
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A relatively weak inequality can be derived very easily ...
See in that given range cos x -sin x >0 ..... (1)
Now first we find that sin x( cos x - sin x) = 1/4 cos^2x -(1/2 cos x- sin x)^2
< 1/4 cos^2x
<1/4 ( in the given range )
so 1/ sin x ( cos x - sin x ) >4
Multiplying Nr . and Dr. by sin x we get
sin x/ sin^2x (cos x - sin x ) >4
but in that range cos x > sin x from (1)
So replacing sin x in the Nr by cos x , we get
cos x/ sin^2x ( cos x- sinx ) >4
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multilpling up and down by (sinx+cosx)
we get cosx(sinx +cosx)/sin2xcos2x>8
assuming it to be true,
we get above as ,
cotx+cot2x>8cos2x (since cos2x >0 in the given interval)
put cotx as t and the above eqution reduces to
t4+t3-7t2+t+8>0
.: it implies that 7t2-t+8<0 (t4 and t3 are +ve)
=> -1<t<8/7
=>-1<cotx<8/7
since 0<x<pi/4
=> 0<cotx<1
thus the interval which we have found is in the given interval.
thus our assumption is true .
and the given inequality is true.