1. We start with the problem: Prove that $\sec^{2}\theta - \cos^{2}\theta = 1$. 2. Recall that $\sec\theta = \frac{1}{\cos\theta}$ by definition. 3. Substitute $\sec^{2}\theta$ with $\frac{1}{\cos^{2}\theta}$: $$\sec^{2}\theta - \cos^{2}\theta = \frac{1}{\cos^{2}\theta} - \cos^{2}\theta$$ 4. To combine the terms, write $\cos^{2}\theta$ as $\frac{\cos^{4}\theta}{\cos^{2}\theta}$ to get common denominator: $$\frac{1}{\cos^{2}\theta} - \frac{\cos^{4}\theta}{\cos^{2}\theta} = \frac{1 - \cos^{4}\theta}{\cos^{2}\theta}$$ 5. Factor numerator $1 - \cos^{4}\theta$ using difference of squares: $$1 - (\cos^{2}\theta)^{2} = (1 - \cos^{2}\theta)(1 + \cos^{2}\theta)$$ 6. Substitute back: $$\frac{(1 - \cos^{2}\theta)(1 + \cos^{2}\theta)}{\cos^{2}\theta}$$ 7. Use the Pythagorean identity $\sin^{2}\theta + \cos^{2}\theta = 1 \Rightarrow 1 - \cos^{2}\theta = \sin^{2}\theta$: $$\frac{\sin^{2}\theta(1 + \cos^{2}\theta)}{\cos^{2}\theta}$$ 8. At this point, notice that the original identity to prove differs from the expression derived in step 7, implying a re-check is necessary. 9. Let's re-express the problem carefully. The problem states $\sec^{2}\theta - \cos^{2}\theta = 1$. 10. Using the identity $\sec^{2}\theta = 1 + \tan^{2}\theta$, the left side becomes: $$1 + \tan^{2}\theta - \cos^{2}\theta$$ 11. Without extra assumptions, this does not simplify directly to 1, indicating original problem perhaps had a typo. 12. Instead, the standard identity is $\sec^{2}\theta - \tan^{2}\theta = 1$. 13. Therefore, the correct identity to prove is: $$\sec^{2}\theta - \tan^{2}\theta = 1$$ which follows directly from the Pythagorean identity. 14. Summary: $\sec^{2}\theta - \cos^{2}\theta = 1$ is not a valid identity. 15. If the intended identity was $\sec^{2}\theta - \tan^{2}\theta = 1$, it's a fundamental trigonometric identity. Final conclusion: The identity $\sec^{2}\theta - \cos^{2}\theta = 1$ is false; please verify the expression to prove.