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Question

The number of integral pairs (x,y) with y>0 satisfying tan1x+cos1y1+y2=sin1310 is 


Solution

tan1x+cos1y1+y2=sin1310
tan1x+tan1(1y)=tan13
tan1(1y)=tan13tan1x
tan1(1y)=tan1(3x1+3x)
y=1+3x3x
y>0x(13,3)
Possible integral values of x are 0,1,2
But, for x=0,y=13 which is not an integer.
For x=1,y=2
For x=2,y=7
Therefore, there are two pairs (x,y) satisfying the given equation.

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