Short Answer 
The problem involves evaluating trigonometric expressions for three pairs of angles (A) and (B). Using the cosine difference identity, ( cos(A-B) ) was calculated for each pair, showing values of approximately 0.866, 0.5, and 0, respectively, while ( cos A – cos B ) provided results of approximately -0.366, -0.985, and 0, confirming the accuracy of the cosine identity.
Step 1: Understand the Problem
The problem involves evaluating trigonometric expressions using the angles (A) and (B). We have three pairs of angles to work with, specifically:
- (A = 60^{circ}, B = 30^{circ})
- (A = 110^{circ}, B = 50^{circ})
- (A = 225^{circ}, B = 135^{circ})
Step 2: Calculate ( cos(A-B) )
Using the cosine formula, ( cos(A – B) = cos A cos B + sin A sin B ), we will evaluate this for each angle pair:
- For (A = 60^{circ}, B = 30^{circ}): ( cos(60^{circ} – 30^{circ}) = cos(30^{circ}) approx 0.866)
- For (A = 110^{circ}, B = 50^{circ}): ( cos(110^{circ} – 50^{circ}) = cos(60^{circ}) = 0.5)
- For (A = 225^{circ}, B = 135^{circ}): ( cos(225^{circ} – 135^{circ}) = cos(90^{circ}) = 0)
Step 3: Evaluate the Expressions
Finally, we will calculate the remaining expressions ( cos A – cos B ) and verify the identity:
- For (A = 60^{circ}, B = 30^{circ}): ( cos 60^{circ} – cos 30^{circ} approx -0.366)
- For (A = 110^{circ}, B = 50^{circ}): ( cos 110^{circ} – cos 50^{circ} approx -0.985)
- For (A = 225^{circ}, B = 135^{circ}): ( cos 225^{circ} – cos 135^{circ} = 0)
- Recheck ( cos A cos B + sin A sin B ) yields same results as ( cos(A-B) ).