simplify the following without the use of a calculator \( \frac{\cos 143^{\circ} \cos 2}{\cos 127} \) \( \sin (-3)) \cdot \sin 217 \cdot \cos 323 \)

We wish to simplify the two expressions  (1) [cos 143° · cos 2°]⁄[cos 127°]  (2) sin(–3°) · sin 217° · cos 323° without using a calculator. (Note: In each case, the angles are measured in degrees.) Below is one way to simplify each expression step‐by‐step. ───────────────────────── Step 1. Simplify Expression (1)  A. Write the expression:   [could also write it as] (cos 143° · cos 2°)⁄(cos 127°)  B. Express cos 143° and cos 127° in terms of acute angles.   • Notice that 143° = 180° – 37° so that    cos 143° = –cos 37°      [since cos(180° – θ) = –cos θ]   • Similarly, 127° = 180° – 53° so that    cos 127° = –cos 53°  C. Substitute these into the expression:    (cos 143° · cos 2°)⁄(cos 127°)    = ((–cos 37°) · cos 2°)⁄(–cos 53°)    = (cos 37° · cos 2°)⁄(cos 53°)  D. Notice that cos 53° is related to sin 37° because 53° and 37° are complementary:   cos 53° = sin (90° – 53°) = sin 37°  E. Thus the expression simplifies to:   (cos 37° · cos 2°)⁄(sin 37°)    = cos 2° · (cos 37°⁄ sin 37°)    = cos 2° · cot 37° That is, one acceptable answer for the first expression is   cos 2° · cot 37°. ───────────────────────── Step 2. Simplify Expression (2)  A. Write the expression:   sin(–3°) · sin 217° · cos 323°  B. Express each factor in terms of a positive acute angle when possible.   • Sinus of a negative angle: sin(–3°) = – sin 3°.   • For sin 217° note that 217° = 180° + 37° so that    sin 217° = – sin 37°    [since sin(180°+θ) = – sin θ]   • For cos 323° note that 323° = 360° – 37° so that    cos 323° = cos 37°     [since cos(360°–θ) = cos θ]  C. Substitute these equivalent forms:    sin(–3°) · sin 217° · cos 323°    = (– sin 3°) · (– sin 37°) · (cos 37°)    = sin 3° · sin 37° · cos 37°  D. It is often useful to combine factors when a product involves sin and cos of the same angle. Note that:   sin 37° · cos 37° = (1/2) sin (2·37°) = (1/2) sin 74°.  E. Thus the product becomes:    sin 3° · (1/2 sin 74°) = (1/2) sin 3° · sin 74°. Either form is acceptable. (That is, you may leave the answer as sin 3° · sin 37° · cos 37° or write it as (1/2) sin 3° · sin 74°.) ───────────────────────── Final Answers (1) (cos 143° · cos 2°)⁄(cos 127°) = cos 2° · cot 37° (2) sin(–3°) · sin 217° · cos 323° = sin 3° · sin 37° · cos 37° = (1/2) sin 3° · sin 74° These forms are fully simplified using standard trigonometric identities and do not require a calculator.