Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>Simplify: \(1 + \tan^2(\tan^{-1} x) - (\sec^2(\sec^{-1} x) - 1)\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental property that inverse trigonometric functions cancel their corresponding trigonometric functions.
<p><strong>Step 1:</strong> Apply the identities $\tan(\tan^{-1} x) = x$ and $\sec(\sec^{-1} x) = x$</p><p>$1 + (\tan(\tan^{-1} x))^2 - (\sec(\sec^{-1} x))^2 + 1$</p><p><strong>Step 2:</strong> Substitute:</p><p>$= 1 + x^2 - x^2 + 1 = 2$</p><p>∴ Answer is <strong>2</strong>.</p>
Correct Answer: 2