Final answer:
To eliminate the parameter in the given parametric equations, we can use the identities sin^2(t) + cos^2(t) = 1 and sin(2t) = 2sin(t)cos(t). From these identities, we can derive the rectangular-coordinate equation 4x = y.
Step-by-step explanation:
To eliminate the parameter in the given parametric equations, we can use the identities sin^2(t) + cos^2(t) = 1 and sin(2t) = 2sin(t)cos(t).
Step-by-step solution:
From the first equation, we have cos^2(t) = x.
From the second equation, we have sin^2(t) = y.
Using the identities, we can rewrite the second equation as 2sin(t)cos(t) = y.
Squaring both sides of the equation, we get (2sin(t)cos(t))^2 = y^2.
Expanding the equation, we have 4sin^2(t)cos^2(t) = y^2.
Substituting the values of sin^2(t) and cos^2(t) from equations 1 and 2, we obtain 4xy = y^2.
Dividing both sides of the equation by y, we get 4x = y.
Therefore, the rectangular-coordinate equation for the curve is 4x = y.