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Question

Inverse circular functions,Principal values of sin1x,cos1x,tan1x.
tan1x+tan1y=tan1x+y1xy, xy<1
π+tan1x+y1xy, xy>1.
(a) If tan1(x+2x)tan14xtan1(x2x)=0
then x=.......... or x=........(xϵR)
(b) If sin1x+sin1 y=2π/3
cos1xcos1 y=π/3
then x=........, y=.......
(c) It tan1y=4tan1x,(|x|<tanπ8), then
express y as an algebraic function of x, Also deduce that tan(π/8) is a root of x46x2+1=0.

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Solution

(a) ±2 combine 1st and 3rd.
x4=4 x2=2 only as 2 is rejected.
(b) x=1/2,y=1. second relation can be written as
(π2sin1x)(π2sin1y)=π3
or sin1y=π/2, sin1X=π/6 etc.
(c) 2tan1x=tan12x1x2
4tan1x=2tan12x1x2
4tan1x=tan12.2x1x21(2x1x2)2
or tan1y=tan14x(1x2)x46x2+1
y=4x(1x2)x46x2+1
Now if x=tanπ8, then
tan1x=π8 or 4tan1x=π2=tan1y
y=tanπ2=
Dr of y=0 or x46x2+1=0
where x=tanπ8 is a root of above.

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