If x = a sec θ cos φ, y = b sec θ sin φ and z = c tan θ, then

Given: x a sec θ cos ϕ

Squaring both sides, we get

x² = a² sec² θ cos²ϕ

and y = b sec θ sin ϕ

Squaring both sides, we get

y² = b² sec² θ sin²ϕ

And z = c tan θ

⇒ z² = c² tan² θ

………(i)

To find:

Consider

= sec² θ cos²ϕ + sec² θ sin²ϕ

= sec² θ (cos²ϕ + sin²ϕ)

= sec² θ [∵ sin²ϕ + cos²ϕ = 1]

= 1 + tan² θ [∵ 1 + tan² θ = sec² θ]