To solve the expression \(\sin^{2.79}(\theta) + \cos^{2.79}(\theta)\), where \(\theta = 79^\circ\), we need to understand that this is not a standard trigonometric identity. The well-known Pythagorean identity states that \(\sin^2(\theta) + \cos^2(\theta) = 1\) for any angle \(\theta\). However, the given expression involves non-integer exponents, which complicates the situation.

Let's break down the expression step-by-step:

1. **Calculate \(\sin(79^\circ)\) and \(\cos(79^\circ)\):**
- \(\sin(79^\circ) \approx 0.9816\)
- \(\cos(79^\circ) \approx 0.1908\)

2. **Raise each value to the power of 2.79:**
- \(\sin^{2.79}(79^\circ) \approx (0.9816)^{2.79}\)
- \(\cos^{2.79}(79^\circ) \approx (0.1908)^{2.79}\)

3. **Compute the powers:**
- \((0.9816)^{2.79} \approx 0.945\)
- \((0.1908)^{2.79} \approx 0.022\)

4. **Add the results:**
- \(0.945 + 0.022 \approx 0.967\)

So, \(\sin^{2.79}(79^\circ) + \cos^{2.79}(79^\circ) \approx 0.967\).

This is an approximate value, and the exact value would require more precise calculations, but this should give you a good estimate.