Question

If x = a(cosθ + θsinθ), y = a(sinθ - θcosθ), prove that d²x/dθ² = a(cosθ - θsinθ), d²y/dθ² = a(sinθ + θcosθ) and d²y/dx² = sec³/aθ.

Answer 1

Given as x = a(cosθ + θsinθ) ...(1)

And

y = a(sinθ - θcosθ) ...(2)

On differentiating eqn (1) with respect to θ

Again, on differentiating eqn (1) with respect to θ

Again, on differentiating with respect to θ

Using eqn. 4 and 5

Again, on differentiating with respect to x 

Put the value in above eqn.