1. **State the problem:** Given that $\tan \theta + \cot \theta = 5$, find the value of $\tan 2\theta + \cot \theta$. 2. **Recall formulas and identities:** - $\cot \theta = \frac{1}{\tan \theta}$. - $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$. 3. **Express the given equation:** $$\tan \theta + \cot \theta = \tan \theta + \frac{1}{\tan \theta} = 5$$ Multiply both sides by $\tan \theta$ (let $t = \tan \theta$): $$t^2 + 1 = 5t$$ 4. **Rewrite as a quadratic equation:** $$t^2 - 5t + 1 = 0$$ 5. **Solve for $t$ using quadratic formula:** $$t = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2}$$ 6. **Calculate $\tan 2\theta$:** $$\tan 2\theta = \frac{2t}{1 - t^2}$$ From step 3, $t^2 = 5t - 1$, so: $$1 - t^2 = 1 - (5t - 1) = 2 - 5t$$ Therefore: $$\tan 2\theta = \frac{2t}{2 - 5t}$$ 7. **Calculate $\tan 2\theta + \cot \theta$:** Recall $\cot \theta = \frac{1}{t}$, so: $$\tan 2\theta + \cot \theta = \frac{2t}{2 - 5t} + \frac{1}{t} = \frac{2t^2}{2 - 5t} + \frac{1}{t}$$ Find common denominator $t(2 - 5t)$: $$= \frac{2t^3 + (2 - 5t)}{t(2 - 5t)}$$ 8. **Simplify numerator:** Using $t^2 = 5t - 1$, find $t^3 = t \cdot t^2 = t(5t - 1) = 5t^2 - t$. Substitute $t^2$ again: $$t^3 = 5(5t - 1) - t = 25t - 5 - t = 24t - 5$$ Now numerator: $$2t^3 + 2 - 5t = 2(24t - 5) + 2 - 5t = 48t - 10 + 2 - 5t = 43t - 8$$ 9. **Final expression:** $$\tan 2\theta + \cot \theta = \frac{43t - 8}{t(2 - 5t)}$$ 10. **Evaluate for each root:** For $t = \frac{5 + \sqrt{21}}{2}$: Calculate numerator and denominator numerically: - Numerator: $43t - 8$ - Denominator: $t(2 - 5t)$ Similarly for $t = \frac{5 - \sqrt{21}}{2}$. 11. **Numerical approximation:** - $\sqrt{21} \approx 4.583$ - For $t_1 = \frac{5 + 4.583}{2} = 4.792$ Numerator: $43 \times 4.792 - 8 = 206.056 - 8 = 198.056$ Denominator: $4.792 \times (2 - 5 \times 4.792) = 4.792 \times (2 - 23.96) = 4.792 \times (-21.96) = -105.25$ Value: $\frac{198.056}{-105.25} \approx -1.88$ - For $t_2 = \frac{5 - 4.583}{2} = 0.2085$ Numerator: $43 \times 0.2085 - 8 = 8.9655 - 8 = 0.9655$ Denominator: $0.2085 \times (2 - 5 \times 0.2085) = 0.2085 \times (2 - 1.0425) = 0.2085 \times 0.9575 = 0.1996$ Value: $\frac{0.9655}{0.1996} \approx 4.84$ 12. **Conclusion:** The possible values of $\tan 2\theta + \cot \theta$ are approximately $-1.88$ or $4.84$ depending on the root chosen for $\tan \theta$. **Final answer:** $$\boxed{\tan 2\theta + \cot \theta \approx -1.88 \text{ or } 4.84}$$