\( \frac { \cos ( 180 ^ { \circ } + \theta ) \cdot \tan ( - \theta ) } { \operatorname { in } ( 360 ^ { \circ } - \theta ) \cdot \tan ( 720 ^ { \circ } + \theta ) } \)

We start with the expression   [cos(180° + θ) · tan(–θ)] / [sin(360° – θ) · tan(720° + θ)] (Note: The notation "in" was interpreted as sin.) Step 1. Simplify the functions using standard trigonometric identities: • cos(180° + θ) = –cos θ • tan(–θ) = –tan θ • sin(360° – θ) = –sin θ   (This is because sin(360° – θ) = sin 360°·cos θ – cos 360°·sin θ = 0·cos θ – 1·sin θ = –sin θ.) • tan(720° + θ) = tan θ   (since tangent has period 180° and 720° is a multiple of 180°) Step 2. Substitute these results into the expression:   Numerator: (–cos θ) · (–tan θ) = cos θ · tan θ   Denominator: (–sin θ) · (tan θ) = –sin θ · tan θ Thus the expression becomes   [cos θ · tan θ] / [–sin θ · tan θ]. Step 3. Cancel the common factor tan θ (assuming tan θ ≠ 0):   = cos θ / (–sin θ)   = –(cos θ/sin θ)   = –cot θ. Therefore, the simplified result is   –cot θ.