1. The problem is to simplify the expression $\csc^2 x + \sec^2 x$. 2. Recall the Pythagorean identities: - $\csc^2 x = 1 + \cot^2 x$ - $\sec^2 x = 1 + \tan^2 x$ 3. Substitute these identities into the expression: $$\csc^2 x + \sec^2 x = (1 + \cot^2 x) + (1 + \tan^2 x)$$ 4. Simplify by combining like terms: $$= 1 + \cot^2 x + 1 + \tan^2 x = 2 + \cot^2 x + \tan^2 x$$ 5. Use the identity $\cot x = \frac{1}{\tan x}$, so $\cot^2 x = \frac{1}{\tan^2 x}$. 6. Rewrite the expression: $$2 + \frac{1}{\tan^2 x} + \tan^2 x$$ 7. Let $t = \tan^2 x$, then the expression becomes: $$2 + t + \frac{1}{t}$$ 8. Combine terms over a common denominator $t$: $$2 + t + \frac{1}{t} = 2 + \frac{t^2 + 1}{t} = \frac{2t + t^2 + 1}{t}$$ 9. The expression cannot be simplified further in a meaningful way without additional context. Final answer: $$\csc^2 x + \sec^2 x = 2 + \cot^2 x + \tan^2 x$$ or equivalently $$2 + \tan^2 x + \frac{1}{\tan^2 x}$$