Finding the value of $x$ when $2^{\frac{1}{\cos^2(x)}}\sqrt{y^2 - y +\frac{1}{2}} \leq 1$

Question

Find the value of $x$ when $2^{\frac{1}{\cos^2(x)}}\sqrt{y^2 - y +\frac{1}{2}} \leq 1$.

I don't know how to start thinking at this question. Please help me. Thank you!

Answer 1

Hint:

We have two unknowns with one inequality.

$$y^2-y+\dfrac12=\left(y-\dfrac12\right)^2+\dfrac12\ge\dfrac12$$

Now for real $x,\sec^2x\ge1\implies2^{\sec^2x}\ge2$

$$\implies2^{\sec^2x}\left(y^2-y+\dfrac12\right)\ge2\cdot\dfrac12$$

What is the intersection with given condtion?