simplify a square root given a sec

Question

simplify √(x²-16) given x=4secθ

Answer 1

Under the square root you have the expression x^2-16=(4secθ)^2-16=16(secθ)^2-16=16((secθ)^2-1)=16(tanθ)^2. Hence the original expression may be rewritten as sqrt(x^2-16)=sqrt(16(tanθ)^2)=4|tanθ|.