1. **Stating the problem:** Simplify and analyze the function $$y = \tan^8 x - 4x^2 \sec x - x^{\cos x}$$. 2. **Recall the definitions and properties:** - $$\tan^8 x = (\tan x)^8$$ means the tangent of $$x$$ raised to the 8th power. - $$\sec x = \frac{1}{\cos x}$$. - $$x^{\cos x}$$ is an exponential expression with base $$x$$ and exponent $$\cos x$$. 3. **Rewrite the function for clarity:** $$y = (\tan x)^8 - 4x^2 \cdot \frac{1}{\cos x} - x^{\cos x}$$ 4. **Important notes:** - The function involves trigonometric powers and an exponential term with a variable exponent. - Simplification depends on the domain of $$x$$ (e.g., $$\cos x \neq 0$$ to avoid division by zero). 5. **No further algebraic simplification is straightforward without specific $$x$$ values.** 6. **Summary:** The function is a combination of a high power of tangent, a rational term involving secant, and an exponential term with a trigonometric exponent. Final answer: $$y = (\tan x)^8 - \frac{4x^2}{\cos x} - x^{\cos x}$$