Question

if x = a sec theta cos phi, y = b sec theta sin phi and z = c tan theta, show (x^2÷a^2)+(y^2÷b^2) - (z^2÷c^2) =1

Answer 1

x = a secθ cosΦ
taking square root both sides,
x² = a² sec²θ cos²Φ
dividing by a² both sides,
x²/a² = sec²θ. cos²Φ --------(1)

similarly,
y²/b² = sec²θ sin²Φ ---------(2)
z²/c² = tan²θ ----------(3)

Now, LHS = x²/a² + y²/b² - z²/c²
= sec²θ.cos²Φ + sec²θ sin²Φ - tan²θ
= sec²θ(cos²Φ + sin²Φ) - tan²θ

we know, sin²x + cos²x = 1
= sec²θ × 1 - tan²θ
= sec²θ - tan²θ = 1 = RHS
as you know, sec²x - tan²x = 1

Answer 2

Answer:

As to prove

Step-by-step explanation:

See in attachment