\( 5.2 .2 \frac { \cos ( - A ) \cdot \tan ( 180 ^ { \circ } - A ) \cdot \sin ( 180 ^ { \circ } - A ) } { \sin ( 360 ^ { \circ } - A ) \cos ( 90 ^ { \circ } - A ) } \)

We wish to simplify   20 · (cos(–A) · tan(180° – A) · sin(180° – A)) ⁄ (sin(360° – A) · cos(90° – A)). It appears that the factors “5.2 .2” indicate a product of 5 · 2 · 2 = 20. Step 1. Rewrite the expression in a more familiar form:   20 · [cos(–A) · tan(180° – A) · sin(180° – A)] ⁄ [sin(360° – A) · cos(90° – A)]. Step 2. Use trigonometric identities for angles:  a. cos(–A) = cos A.  b. tan(180° – A):    Recall: tan(θ) = sinθ/cosθ.    Since sin(180° – A) = sin A and cos(180° – A) = –cos A,    tan(180° – A) = (sin A)/(–cos A) = –(sin A/cos A) = –tan A.  c. sin(180° – A) = sin A.  d. sin(360° – A):    Using the periodicity and symmetry of sine, sin(360° – A) = – sin A.  e. cos(90° – A) = sin A. Step 3. Substitute these results into the expression:   20 · [cos A · (–tan A) · sin A] ⁄ [(– sin A) · sin A]. Step 4. Simplify the negatives:   The numerator has a factor of – and the denominator also is – sin A · sin A.   Cancel the negative signs:    20 · [cos A · tan A · sin A] ⁄ [sin A · sin A]. Step 5. Express tan A as sin A/cos A:   20 · [cos A · (sin A/cos A) · sin A] ⁄ [sin² A]. Step 6. Cancel the common factors:   The cos A in the numerator cancels with cos A in the denominator inside tan A, and one factor of sin A cancels one sin A in the denominator:    20 · [sin A · sin A/(sin² A)] = 20 · 1 = 20. Thus, the expression simplifies to 20.