Problem statement: If xsin^3theta ycos^3theta = sintheta.costheta and xsintheta = ycostheta , prove that x^2 y^2 =1.

Solution:
We need to prove that x^2 y^2 =1. From the given equations, we can manipulate them to get x^2 y^2 =1.

Manipulating the given equations:
We have, xsin^3theta ycos^3theta = sintheta.costheta and xsintheta = ycostheta.

Squaring both the equations, we get:

x^2sin^6theta + 2xy^2sin^3theta.cos^3theta + y^2cos^6theta = sin^2theta.cos^2theta

x^2sin^2theta = y^2cos^2theta

Multiplying the above two equations, we get:

x^2sin^2theta.y^2cos^2theta = sin^2theta.cos^2theta

Substituting the value of x^2sin^2theta from the second equation, we get:

y^2cos^2theta.y^2cos^2theta = sin^2theta.cos^2theta

Simplifying the above equation, we get:

y^4cos^4theta = sin^2theta.cos^2theta

Taking the square root on both sides, we get:

y^2cos^2theta = sintheta.costheta

Substituting the value of x^2sin^2theta from the second equation, we get:

x^2sin^2theta.cos^2theta = sintheta.costheta

Simplifying the above equation, we get:

x^2 = 1/cos^2theta

y^2 = 1/sin^2theta

Multiplying the above two equations, we get:

x^2 y^2 = 1

Therefore, we have proved that x^2 y^2 = 1.