Questions: Find the exact value of sec(2 tan^(-1) (5/12)).

Solution Steps

To find the exact value of \(\sec \left(2 \tan ^{-1} \frac{5}{12}\right)\), we can use the double-angle identity for tangent and the relationship between secant and cosine.

1. Let \(\theta = \tan^{-1} \frac{5}{12}\). Then, \(\tan \theta = \frac{5}{12}\).
2. Use the double-angle identity for tangent: \(\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).
3. Calculate \(\tan(2\theta)\) using the value of \(\tan \theta\).
4. Use the identity \(\sec(2\theta) = \frac{1}{\cos(2\theta)}\) and the Pythagorean identity to find \(\cos(2\theta)\).
5. Finally, compute \(\sec(2\theta)\).

Let \(\theta = \tan^{-1} \frac{5}{12}\). Thus, we have: \[ \tan \theta = \frac{5}{12} \]

Using the double-angle identity for tangent: \[ \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \] Substituting \(\tan \theta\): \[ \tan(2\theta) = \frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2} = \frac{\frac{10}{12}}{1 - \frac{25}{144}} = \frac{\frac{10}{12}}{\frac{119}{144}} = \frac{10 \cdot 144}{12 \cdot 119} = \frac{120}{119} \approx 1.0084 \]

Using the identity \(\sec(2\theta) = \sqrt{1 + \tan^2(2\theta)}\): \[ \sec(2\theta) = \sqrt{1 + \left(\tan(2\theta)\right)^2} = \sqrt{1 + \left(1.0084\right)^2} \approx \sqrt{1 + 1.0170} \approx \sqrt{2.0170} \approx 1.4202 \]

Final Answer