6 Simplify the expression 4pts sec tan Vx21 x 1 Solutionsect

6. Simplify the expression. (4pts) sec (tan Vx2-1) (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

hypotenuse(sqrt(x²-1))²+ 1²

hypotenuse² = x²-1+1

hypotenuse²= x²

hypotenuse = x

Sec theta= hypotenuse/ adjacent

Therefore sec t= x/1

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

sec(tan^-1(sqrt(x²-1))) = x