\( \left. \begin{array} { l } { \cos 80 ^ { \circ } + \cos 40 ^ { \circ } = \cos 20 ^ { \circ } } \\ { \sin 75 ^ { \circ } - \cos 105 ^ { \circ } - \sin 15 ^ { \circ } - \cos 15 ^ { \circ } = 0 } \\ { \tan ( A + 45 ^ { \circ } ) = \frac { 1 + \tan A } { 1 - \tan A } } \end{array} \right. \)

Check the equality by following steps: - step0: Check the equality is true or false: \(\cos\left(80^{\circ}\right)+\cos\left(40^{\circ}\right)-\cos\left(20^{\circ}\right)=0\) - step1: Simplify: \(0=0\) - step2: Check the equality: \(\textrm{true}\) Solve the equation \( \sin 75 ^ { \circ } - \cos 105 ^ { \circ } - \sin 15 ^ { \circ } - \cos 15 ^ { \circ } = 0 \). Check the equality by following steps: - step0: Check the equality is true or false: \(\sin\left(75^{\circ}\right)-\cos\left(105^{\circ}\right)-\sin\left(15^{\circ}\right)-\cos\left(15^{\circ}\right)=0\) - step1: Simplify: \(0=0\) - step2: Check the equality: \(\textrm{true}\) Let's analyze the three equations step by step. 1. **First Equation:** \[ \cos 80^{\circ} + \cos 40^{\circ} = \cos 20^{\circ} \] This equation is true. The left-hand side simplifies to the right-hand side, confirming that this equation holds for all real values. 2. **Second Equation:** \[ \sin 75^{\circ} - \cos 105^{\circ} - \sin 15^{\circ} - \cos 15^{\circ} = 0 \] This equation is also true. The left-hand side simplifies to zero, confirming that this equation holds for all real values. 3. **Third Equation:** \[ \tan(A + 45^{\circ}) = \frac{1 + \tan A}{1 - \tan A} \] This equation is a known identity in trigonometry, which states that: \[ \tan(A + 45^{\circ}) = \frac{\tan A + 1}{1 - \tan A} \] However, it seems that the calculation for this equation was unsupported. In summary: - The first two equations are true for all real values. - The third equation is a trigonometric identity, but we couldn't compute a specific solution due to unsupported calculations. If you need further assistance or specific values for \( A \) in the third equation, please let me know!