\( \tan ( x + y ) + \sec ( x - y ) = 2 \)

To solve the equation \( \tan ( x + y ) + \sec ( x - y ) = 2 \), we can start by rewriting the trigonometric functions in terms of sine and cosine. Recall that: \[ \tan(x+y) = \frac{\sin(x+y)}{\cos(x+y)} \] and \[ \sec(x-y) = \frac{1}{\cos(x-y)}. \] Substituting these into the equation gives: \[ \frac{\sin(x+y)}{\cos(x+y)} + \frac{1}{\cos(x-y)} = 2. \] To simplify the equation, we can multiply through by \(\cos(x+y) \cos(x-y)\) (assuming \(\cos(x+y) \neq 0\) and \(\cos(x-y) \neq 0\)): \[ \sin(x+y) \cos(x-y) + \cos(x+y) = 2 \cos(x+y) \cos(x-y). \] Rearranging this, we have: \[ \sin(x+y) \cos(x-y) + \cos(x+y) - 2 \cos(x+y) \cos(x-y) = 0. \] This can be rewritten as: \[ \sin(x+y) \cos(x-y) + \cos(x+y)(1 - 2 \cos(x-y)) = 0. \] Now, let's analyze the original equation \( \tan(x+y) + \sec(x-y) = 2 \) for specific values of \(x\) and \(y\). ### Case 1: \(x = y = 0\) \[ \tan(0 + 0) + \sec(0 - 0) = \tan(0) + \sec(0) = 0 + 1 = 1 \quad (\text{not a solution}). \] ### Case 2: \(x = y = \frac{\pi}{4}\) \[ \tan\left(\frac{\pi}{4} + \frac{\pi}{4}\right) + \sec\left(\frac{\pi}{4} - \frac{\pi}{4}\right) = \tan\left(\frac{\pi}{2}\right) + \sec(0) = \text{undefined} + 1 \quad (\text{not a solution}). \] ### Case 3: \(x = y = \frac{\pi}{6}\) \[ \tan\left(\frac{\pi}{6} + \frac{\pi}{6}\right) + \sec\left(\frac{\pi}{6} - \frac{\pi}{6}\right) = \tan\left(\frac{\pi}{3}\right) + \sec(0) = \sqrt{3} + 1 \quad (\text{not a solution}). \] ### Case 4: \(x = y = \frac{\pi}{3}\) \[ \tan\left(\frac{\pi}{3} + \frac{\pi}{3}\right) + \sec\left(\frac{\pi}{3} - \frac{\pi}{3}\right) = \tan\left(\frac{2\pi}{3}\right) + \sec(0) = -\sqrt{3} + 1 \quad (\text{not a solution}). \] ### General Approach To find solutions, we can also consider the possibility of \(x + y\) and \(x - y\) being specific angles that yield simple values for \(\tan\) and \(\sec\). ### Conclusion The equation \( \tan(x+y) + \sec(x-y) = 2 \) can be complex to solve analytically without specific values or constraints on \(x\) and \(y\). To find specific solutions, numerical methods or graphing might be necessary, or we can look for specific angles that satisfy the equation. If you have any specific constraints or values for \(x\) and \(y\), please provide them for a more targeted solution.