Minimizing $4x^2-4x|\sin\theta|-\cos^2\theta$

Question

Find the minimum value of $$4x^2-4x|\sin\theta|-\cos^2\theta$$ over real $x$ and $\theta$.

Answer 1

Given, $4x^2-4x|\sin\theta|-\cos^2\theta$ Now, adding and subtracting $\sin^2\theta$ The equation becomes $(4x^2+\sin^2\theta-4x|\sin\theta|)-\sin^2\theta-\cos^2\theta$

The bracketed portion of the Function is of the form $(a-b)^2$ . And we know the equality of $\sin^2\theta+\cos^2\theta=1$ . So, reducing the terms. $(2x-|\sin\theta|)^2-(\sin^2\theta+\cos^2\theta)$ This reduces to $(2x-|\sin\theta|)^2-1$ Now, we know, the minimum value of $(2x-|\sin\theta|)^2$ can be only $0$, irrespective of $x$ and $\theta$ . So, by applying that we get that the minimum value of the equation is $-1$ .