1. **State the problem:** Evaluate the expression $$\cos\left(-\frac{\pi}{4}\right) + \sin\left(-\frac{7\pi}{6}\right)$$. 2. **Recall the relevant formulas and properties:** - Cosine is an even function, so $$\cos(-x) = \cos(x)$$. - Sine is an odd function, so $$\sin(-x) = -\sin(x)$$. 3. **Apply these properties:** $$\cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)$$ $$\sin\left(-\frac{7\pi}{6}\right) = -\sin\left(\frac{7\pi}{6}\right)$$ 4. **Evaluate each trigonometric function:** - $$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$ - $$\sin\left(\frac{7\pi}{6}\right)$$ is in the third quadrant where sine is negative, and its reference angle is $$\frac{\pi}{6}$$, so: $$\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$$ 5. **Substitute back:** $$\cos\left(-\frac{\pi}{4}\right) + \sin\left(-\frac{7\pi}{6}\right) = \frac{\sqrt{2}}{2} + \left(-\left(-\frac{1}{2}\right)\right) = \frac{\sqrt{2}}{2} + \frac{1}{2}$$ 6. **Final answer:** $$\boxed{\frac{\sqrt{2}}{2} + \frac{1}{2}}$$