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Question

Find the value of tan12[sin12x1+x2+cos11y21+y2]
|x|<1,y>0 and xy<1

A
x+y1xy
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B
π+x+y1xy
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C
1xyx+y
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D
π+1xyx+y
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Solution

The correct option is A x+y1xy
Let x=tanθ, then θ=tan1x.
sin12x1+x2=sin1(2tanθ1+tan2θ)
=sin1(sin2θ)=2θ=2tan1x
Let y=tanϕ,Thenϕ=tan1y
cos11y21+y2=cos1(1tan2ϕ1+tan2ϕ)
=cos1(cos2ϕ)=2ϕ=2tan1y
tan12[sin12x1+x2+cos11y21+y2]
=tan12[2tan1x+2tan1y]
=tan[tan1x+tan1y]
=tan[tan1(x+y1xy)]=x+y1xy

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