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]


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