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Question

Write each of the following in the simplest form:

(i)
cot-1ax2-a2, x >a
(ii) tan-1x+1+x2, xR

(iii) tan-11+x2-x, xR

(iv) tan-11+x2-1x, x0

(v) tan-11+x2+1x, x0

(vi) tan-1a-xa+x,-a<x<a

(vii) tan-1xa+a2-x2,-a<x<a

(viii) sin-1x+1-x22,-1<x<1

(ix) sin-11+x+1-x2, 0<x<1

(x) sin2 tan-11-x1+x

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Solution

(i) Let x=asecθ
Now,

cot-1ax2-a2=cot-1aa2sec2θ-a2=cot-1aatan2θ=cot-1cotθ=θ =sec-1xa

(ii) Let x=cotθ
Now,
tan-1x+1+x2=tan-1cotθ+1+cot2θ=tan-1cotθ+cosecθ =tan-1cosθ+1sinθ=tan-12cos2θ22sinθ2cosθ2=tan-1cotθ2=tan-1tanπ2-θ2 =π2-θ2=π2-cot-1x2



(iii) Let x=cotθ
Now,

tan-11+x2-x=tan-11+cot2θ-cotθ=tan-1cosecθ-cotθ=tan-11-cosθsinθ=tan-12sin2θ22sinθ2cosθ2=tan-1tanθ2=θ2=cot-1x2

(iv) Let x=tanθ
Now,

tan-11+x2-1x=tan-11+tan2θ-1tanθ=tan-1sec2θ-1tanθ =tan-1secθ-1tanθ=tan-11-cosθsinθ=tan-12sin2θ22sinθ2cosθ2=tan-1tanθ2=θ2=tan-1x2

(v) Let x=tanθ
Now,

tan-11+x2+1x=tan-11+tan2θ+1tanθ=tan-1sec2θ+1tanθ =tan-1secθ+1tanθ=tan-1cosθ+1sinθ=tan-12cos2θ22sinθ2cosθ2=tan-1cotθ2=tan-1tanπ2-θ2 =π2-θ2=π2-tan-1x2


(vi) Let x=acosθ
Now,

tan-1a-xa+x=tan-1a-acosθa+acosθ=tan-11-cosθ1+cosθ=tan-12sin2θ22cos2θ2=tan-1tanθ2=θ2 =12cos-1xa
tan-1a-xa+x=cos-1xa2

(vii) Let x=asinθ
Now,

tan-1xa+a2-x2=tan-1asinθa+a2-a2cos2θ=tan-1asinθa+acos2θ=tan-1sinθ1+cosθ=tan-12sinθ2cosθ22cos2θ2=tan-1tanθ2 =θ2=12sin-1xa


(viii) Let x=sinθ
Now,

sin-1x+1-x22=sin-1sinθ+1-sin2θ2=sin-1sinθ+cosθ2=sin-112sinθ+12cosθ=sin-1cosπ4sinθ+sinπ4cosθ=sin-1sinθ+π4 =θ+π4=π4+sin-1x
sin-1x+1-x22=cos-1x+π4

(ix) Let x=cosθ
Now,
sin-11+x+1-x2=sin-11+cosθ+1-cosθ2=sin-12cos2θ2+2sin2θ22=sin-1cosθ2+sinθ22=sin-112sinθ2+12cosθ2=sin-1sinθ2+π4 =θ2+π4=cos-1x2+π4
sin-11+x+1-x2=cos-1x2+π4

(x) Let x=cosθ
Now,

sin2tan-11-x1+x=sin2tan-11-cosθ1+cosθ=sin2tan-12sin2θ22cos2θ2=sin2tan-1tanθ2=sinθ=sincos-1x =sinsin-11-x2=1-x2

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