Let x = 3 sec theta!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question

Find the sqrt [x^2 -9]/x^2?

Answer 1

x = 3secθ

Take square to each side

x² = 9sec²θ

Recall : Trogonometry formulas sec²θ - tan²θ = 1

Substitute sec²θ = 1 + tan²θ in the x² = 9sec²θ

x² = 9(1 + tan²θ)

Recall : Distributive property a(b + c) = ab + ac

x² = 9 + 9tan²θ

Subtract 9 from each side

x² - 9 = 9tan²θ

Take square root to each side

√[x² - 9] = 3tanθ

√[x² - 9] / x² = 3tanθ / 9sec²θ

√[x² - 9] / x² = tanθ / 3sec²θ

√[x² - 9] / x^2 = tanθ*cos^2θ / 3

Simplify

√[x² - 9] / x² = (sinθ / cosθ)*cos²θ / 3

√[x² - 9] / x² = (sinθ * cosθ) / 3

√[x² - 9] / x² = (sin2θ) / 6.