1. \( \frac{\cot \theta+2 \cos \theta}{\cos \theta-\operatorname{sen} \theta}=\sec \theta+2 \tan \theta \)

To solve the equation \( \frac{\cot \theta + 2 \cos \theta}{\cos \theta - \sin \theta} = \sec \theta + 2 \tan \theta \), we can start by rewriting the trigonometric functions in terms of sine and cosine. Recall the relationships: - \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) - \( \sec \theta = \frac{1}{\cos \theta} \) - \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) Let’s rewrite the left hand side: \[ \frac{\frac{\cos \theta}{\sin \theta} + 2 \cos \theta}{\cos \theta - \sin \theta} = \frac{\frac{\cos \theta + 2 \cos \theta \sin \theta}{\sin \theta}}{\cos \theta - \sin \theta} \] Simplifying the left: \[ \frac{\cos \theta + 2 \cos \theta \sin \theta}{\sin \theta(\cos \theta - \sin \theta)} \] Now, simplifying the right hand side: \[ \sec \theta + 2 \tan \theta = \frac{1}{\cos \theta} + 2 \cdot \frac{\sin \theta}{\cos \theta} = \frac{1 + 2 \sin \theta}{\cos \theta} \] Now we can equate the two sides: \[ \frac{\cos \theta + 2 \cos \theta \sin \theta}{\sin \theta(\cos \theta - \sin \theta)} = \frac{1 + 2 \sin \theta}{\cos \theta} \] Cross-multiplying to eliminate the fractions (assuming \( \sin \theta \neq 0 \) and \(\cos \theta \neq 0\)): \[ (\cos \theta + 2 \cos \theta \sin \theta)\cos \theta = (1 + 2 \sin \theta) \sin \theta(\cos \theta - \sin \theta) \] This will lead to a solvable equation. Expand and simplify both sides to find the values for \( \theta \). Depending on specific angles and using the identities related to angles, we can derive solutions manually or employ a calculator for approximate values if needed. You should check for possible restrictions in the domains of the variables to avoid undefined situations, especially because secant and tangent functions can introduce vertical asymptotes in certain angles (like \(\theta = 90^\circ\) or \(\theta = 270^\circ\)). Thus, after simplifying, you can graph or use numerical techniques to solve for \( \theta \) effectively.