1. **State the problem:** Simplify the expression $$(\tanh x - 1)(\tanh x + 1) - 2 \tanh^2 x + \sec^2 x$$ using trigonometric identities. 2. **Recall relevant identities:** - Difference of squares: $$(a - b)(a + b) = a^2 - b^2$$ - Hyperbolic tangent squared identity: $$\tanh^2 x = \frac{\sinh^2 x}{\cosh^2 x}$$ - Pythagorean identity for secant: $$\sec^2 x = 1 + \tan^2 x$$ 3. **Simplify the first part:** $$(\tanh x - 1)(\tanh x + 1) = \tanh^2 x - 1$$ 4. **Rewrite the entire expression:** $$\tanh^2 x - 1 - 2 \tanh^2 x + \sec^2 x = - \tanh^2 x - 1 + \sec^2 x$$ 5. **Express $\tanh^2 x$ in terms of $\tan^2 x$:** Note that $\tanh x$ is hyperbolic tangent, and $\tan x$ is tangent; they are different functions. Since the expression mixes $\tanh$ and $\sec$, which is related to $\tan$, we keep them separate. 6. **Use the identity for $\sec^2 x$:** $$\sec^2 x = 1 + \tan^2 x$$ 7. **Substitute into the expression:** $$- \tanh^2 x - 1 + 1 + \tan^2 x = - \tanh^2 x + \tan^2 x$$ 8. **Final simplified form:** $$\tan^2 x - \tanh^2 x$$ **Answer:** The simplified expression is $$\tan^2 x - \tanh^2 x$$.