The value of sinh-1x1-x2 is
tanh-1x
coth-1x
sinh-12x
cosh-12x
Explanation for the correct option
The given trigonometric expression: sinh-1x1-x2.
Let us assume that, sinh-1x1-x2=y
⇒sinhy=x1-x2
It is known that, cosh2(θ)-sinh2(θ)=1.
Thus, cosh2y-sinh2y=1.
⇒cosh2y-x1-x22=1⇒cosh2y-x21-x2=1⇒cosh2y=1+x21-x2⇒cosh2y=1-x2+x21-x2⇒cosh2y=11-x2⇒coshy=11-x2
Since, tanhy=sinhycoshy
⇒tanhy=x1-x211-x2⇒tanhy=x⇒y=tanh-1x⇒sinh-1x1-x2=tanh-1x[∵sinh-1x1-x2=y]
Therefore, the value of sinh-1x1-x2 is tanh-1x.
Hence, the correct option is (A).