How to simplify #tan(sec^(-1)(x))# ?

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Answer 1 · 2018-08-02T23:42:24

#tan(sec^(-1)(x))=sqrt(x^2-1)#

#tan(sec^(-1)(x))#

#=tan(cos^(-1)(1/x))#

let #y=cos^(-1)(1/x)#
#x=1/cos(y)#

#x^2=1/cos(y)^2#

#x^2-1=(cancel(1-cos(y)^2)^(=sin(y)^2))/(cos(y)^2)#

#x^2-1=tan(y)^2#

#sqrt(x^2-1)=tan(y)=tan(sec^(-1)(x))#

\0/ Here's our answer !

Answer 2 · 2018-08-03T02:20:30

#color(crimson)(tan(sec^-1(x)) =sqrt ( x^2 -1)#

#tan (sec^-1 x)#

Let #sec ^-1 x = y#

#x = sec y#

#x^2 = sec^2 y#

#x^2 = 1 + tan^2 y, color(brown)(sec^2 y = 1 + tan^2 y, " Identity"#

#x^2 - 1= tan^2 y#

#tan y = sqrt(x^2 - 1)#

#color(crimson)(tan(sec^-1(x)) = sqrt(x^2 -1)#