Question #e9266

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Answer 1 · 2017-08-29T16:06:46

After using u= #cot^(-1)(sqrt(1+x)/sqrt(1-x))# transform,

#cotu=sqrt(1+x)/sqrt(1-x)#

#(cotu)^2=(1+x)/(1-x)#

#(1-x)*(cotu)^2=1+x#

#(cotu)^2-x*(cotu)^2=1+x#

#x*[(cotu)^2+1]=(cotu)^2-1#

#x*(cscu)^2=(cscu)^2*[(cosu)^2-(sinu)^2]#

#x=(cosu)^2-(sinu)^2#

#x=cos2u#

Hence,

#y=(sin[cot^(-1)(sqrt(1+x)/sqrt(1-x))])^2#

=#(sinu)^2#

=#1/2*2(sinu)^2#

=#1/2*(1-cos2u)#

=#(1-x)/2#

1) I used #u=cot^(-1)(sqrt(1+x)/sqrt(1-x))# transform for finding x in terms of u. Finally I found #x=cos2u#.

2) I rewrote #(sinu)^2# in terms of #cos2u#. Finally I found #(sin[cot^(-1)(sqrt(1+x)/sqrt(1-x))])^2# as #(1-x)/2#.