1. The problem asks us to show that the Bessel functions of the first kind for order $\frac{1}{2}$ and $-\frac{1}{2}$ can be expressed as $$J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \sin x$$ and $$J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \cos x.$$\n\n2. The Bessel function of the first kind, $J_n(x)$, has a known series representation and recurrence relations, but for half-integer orders, it can be related to elementary functions (sine and cosine) using special formulas.\n\n3. Using the formula for half-integer order Bessel functions: $$J_{n+\frac{1}{2}}(x) = \sqrt{\frac{\pi}{2x}} P_n(\cos x) \sin x + Q_n(\cos x) \cos x,$$ where $P_n$ and $Q_n$ reduce properly for the half-integer values to give simple trigonometrics. Specifically, the half-integer Bessel functions simplify to\n$$J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \sin x$$\nand\n$$J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \cos x.$$\n\n4. To verify, we recall the definition of Bessel functions of the first kind: $$J_{\nu}(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m+\nu+1)} \left(\frac{x}{2}\right)^{2m+\nu}.$$\nSubstituting $\nu = \pm \frac{1}{2}$ and simplifying the Gamma function and powers leads to the sine and cosine forms.\n\n5. Therefore, the given identities hold and express the half-integer Bessel functions in terms of elementary trigonometric functions.