Question

If a = x sinθ + y cosθc and b = x cosθ – y sinθi then prove that, d3x/dθ3 . d2y/dθ2 – d2y/dθ2. d3y/dθ3 = a2 + b2.

Answer 1

Given x sinθ + y cosθ = a

xcosθ – y sinθ = b

On solving, we get,

x = asinθ + bcosθ

y = acosθ – bsinθ

Now, put a = rsinϕ, b = rcosϕ