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

2 Answers
Aug 2, 2018

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

Explanation:

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 !

Aug 3, 2018

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

Explanation:

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)