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Question

If sinxsiny=12,cosxcosy=32 where x,y(0,π2) and the value of tan(x+y) is k, then [k]=
(where [.] denotes greatest integer function)

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Solution

sinxsiny=12, cosxcosy=32
tanxtany=13 ...(1)

Now,
tan(x+y)=tanx+tany1tanxtany
From equation (1), we get
=4tanx13tan2x

Also,
siny=2sinx, cosy=23cosx
sin2y+cos2y=14sin2x+49cos2x=136sin2x+4cos2x=936tan2x+4=9sec2x36tan2x+4=9(1+tan2x)27tan2x=5tanx=533 (x(0,π/2))tan(x+y)=453311527=15k=15[k]=3

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