Given Equations
We start with the two equations provided:
- X sin^3 A + Y cos^3 A = sin A cos A
- X sin A = Y cos A
Substituting Values
From the second equation, we can express Y in terms of X:
- Y = (X sin A) / (cos A)
Now, substitute this expression for Y into the first equation:
- X sin^3 A + ((X sin A) / (cos A)) cos^3 A = sin A cos A
Simplifying the Equation
This simplifies as follows:
- X sin^3 A + X sin A (cos^2 A) = sin A cos A
Factor out X sin A:
- X sin A (sin^2 A + cos^2 A) = sin A cos A
Since sin^2 A + cos^2 A = 1, the equation reduces to:
- X sin A = sin A cos A
Dividing Both Sides
Assuming sin A ≠ 0, we can safely divide both sides by sin A:
- X = cos A
Finding Y in Terms of X
Now, substitute X back into the equation for Y:
- Y = (X sin A) / (cos A) = (cos A sin A) / (cos A) = sin A
Calculating X² and Y²
Now we can express X and Y:
- X = cos A
- Y = sin A
To find X² + Y²:
- X² + Y² = (cos A)² + (sin A)² = 1
Conclusion
Thus, we conclude:
- X² + Y² = 1, which proves that X² Y² = 1, given that X and Y are defined in terms of sin A and cos A.