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Question

Find all positive integral solution of the equation tan1x+cot1y=tan13

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Solution

We have to find values of x and y which are positive as well as integer
Given:tan1x+cot1y=tan13
tan1x+tan11y=tan13 where y>0
tan1⎜ ⎜ ⎜x+1y1xy⎟ ⎟ ⎟=tan13
tantan1⎜ ⎜ ⎜x+1y1xy⎟ ⎟ ⎟=tantan13
x+1y1xy=3
3x+3y=1xy
3x+1=y(3x)
y=3x+13x
when x is positive, numerator is positive for y to be positive denominator must be positive
i.e., 3x>0 or x<3x=1,2
for x=1y=2
for x=2y=7

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