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Question

If cos2(x+y)cos2(xy)4sinxsiny=limx0+sgn(sgn(sgn x)), where sgn is signum function, then which of the following statements is/are correct? 

 
  1. x=nπ and y=mπ, where n,m are both even integers
  2. x=nπ and y=mπ, where n is even and m is an odd integer.
  3. x=nπ and y=mπ, where n,m are both odd integers
  4. x=nπ and y=mπ, where n is odd and m is an even integer.


Solution

The correct options are
A x=nπ and y=mπ, where n,m are both even integers
C x=nπ and y=mπ, where n,m are both odd integers
cos2(x+y)cos2(xy)4sinxsiny=limx0+sgn(sgn(sgn x))

cos2(x+y)cos2(xy)2(cos(x+y)cos(xy))=1

cos(x+y)+cos(xy)2=1

cosxcosy=1

Now we know that,
1cosx1, and 1cosy1
So, cosx=1 and cosy=1
or cosx=1 and cosy=1
x=2nπ and y=2mπ (n,mZ)
or x=(2n+1)π and y=(2m+1)π (n,mZ)

x=nπ and y=mπ where n,m are both even integers or both odd integers.

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