Demonstrate that $x(t)$ and $\hat{x}(t)$ are orthogonal
You have to use the following form of Parseval's identity:

$$\int_{-\infty}^{\infty}x(t)y^*(t)dt=\int_{-\infty}^{\infty}X(f)Y^*(f)df\tag{1}$$

where $^*$ denotes complex conjugation, and where $X(f)$ and $Y(f)$ are the Fourier transforms of $x(t)$ and $y(t)$, respectively. With $y(t)=\hat{x}(t)$ and $Y(f)=-j\,\textrm{sgn}(f)X(f)$, we get from $(1)$

$$\int_{-\infty}^{\infty}x(t)\hat{x}^*(t)dt=j\int_{-\infty}^{\infty}|X(f)|^2\textrm{sgn}(f)df\tag{2}$$

For real-valued $x(t)$, $|X(f)|^2$ is an even function, and, consequently, the integrand on the right-hand side of $(2)$ is odd. This implies that the integral on the right-hand side of $(2)$ equals zero if it is interpreted as a Cauchy principal value.

Note that we've used the fact that $x(t)$ is real-valued. For complex-valued $x(t)$, the integrals in $(2)$ are generally not zero.