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]