Proof of the stiffness of the horizontal component of a spring

[jonathan308]

Homework Statement

Suppose a spring with forces F=k*z. Find the equivalent stiffness of a horizontal spring, where the spring and its horizontal equivalent is separated by theta degrees. The horizontal equivalent presumably will have a force F(x)=k(x)*x

Homework Equations

F = F(x)*cos(theta)
x = z*cos(theta)

The Attempt at a Solution

I have try to substitute F=k*z, F = F(x)*cos(theta) and x=z*cos(theta) into the equation F(x)=k(x)*x, but i only end up with a equation k = k(x)*cos^2(theta). The answer should be k(x)=k*cos^2(theta) and it only makes sense if the horizontal equivalent of stiffness is smaller than k. Please help and point out where I have made a mistake. thanks :)

 

[Lnewqban]

Welcome!
Could you show any schematic?