To find dy/dx, we need to calculate the derivative of y with respect to x.
Given y = cos(t) and x = sin(t), we can express cos(t) in terms of x:
cos(t) = cos(arcsin(x))
To find dy/dx, we will differentiate both sides of the equation with respect to x:
d/dx (cos(t)) = d/dx (cos(arcsin(x)))
Using the chain rule, the right side becomes:
-sin(arcsin(x)) * d/dx (arcsin(x))
We know that sin(arcsin(x)) = x, so we can simplify the equation to:
-d/dx (arcsin(x))
The derivative of arcsin(x) with respect to x is:
d/dx (arcsin(x)) = 1 / sqrt(1 - x^2)
Therefore, dy/dx = -1 / sqrt(1 - x^2)
So, dy/dx = -1 / sqrt(1 - sin^2(t))
Notice that if we substitute sin(t) into the equation, we get:
dy/dx = -1 / sqrt(1 - sin^2(t))
Thus, dy/dx = -1 / sqrt(1 - x^2) for all values of x.