Simplify the expression sec tan1 squareroot x2 1 x 1Soluti

Simplify the expression. sec (tan^-1 squareroot x^2 - 1) (x > 1)

Solution

sec(tan^-1(sqrt(x²-1)))

let tan^-1(sqrt(x²-1)) = t

taking tan on both sides

tan(tan^-1(sqrt(x²-1)))= tan t

sqrt(x²-1) = tan t

tan t = sqrt(x²-1)/1

tan theta=opposite/adjacent

Therefore opposite=sqrt(x²-1) , adjacent=1

Using pythagoras theorem

hypotenuse²=opposite²+ adjacent²

hypotenuse²= (sqrt(x²-1))² + 1²

hypotenuse²= x²-1 + 1

hypotenuse²= x²

hypotenuse = x

sec theta= hypotenuse/ adjacent

therefore

sec t= x/1

sec t= x

Substituting back tan^-1(sqrt(x²-1)) for t

sec(tan^-1(sqrt(x²-1)))= x