If x = a tan(theta) and y = b sec(theta), then we can express the relationship between x and y as follows:

If x = a tan(theta) and y = b sec(theta), then we can express the relationship between x and y as follows:

Given:

x = a tan(theta)

y = b sec(theta)

We know that:

sec(theta) = 1 / cos(theta)

tan(theta) = sin(theta) / cos(theta)

Substituting the values of sec(theta) and tan(theta), we get:

x = a (sin(theta) / cos(theta))

y = b (1 / cos(theta))

Dividing both sides of the second equation by cos(theta), we get:

y / cos(theta) = b

Substituting y / cos(theta) in the first equation, we get:

x = a (sin(theta) / (y / b))

Cross-multiplying, we get:

x = (a * b * sin(theta)) / y

Therefore, the relationship between x and y can be expressed as:

x = (a * b * sin(theta)) / y

This equation shows that x is directly proportional to the product of a and b, and inversely proportional to y. It also involves the sine of the angle theta.

Note: The variables a and b represent constant values, while theta is the variable angle.