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Given:
sin A = 4/5 and cos B = 5/13
We know that cos A = √(1 – sin2 A) and sin B = √(1 – cos2 B), where 0
So let us find the value of sin A and cos B
cos A = √(1 – sin2 A)
= √(1 – (4/5)2)
= √(1 – (16/25))
= √((25 – 16)/25)
= √(9/25)
= 3/5
sin B = √(1 – cos2 B)
= √(1 – (5/13)2)
= √(1 – (25/169))
= √(169 – 25)/169)
= √(144/169)
= 12/13

(i) sin (A + B)
We know that sin (A +B) = sin A cos B + cos A sin B
So,
sin (A +B) = sin A cos B + cos A sin B
= 4/5 × 5/13 + 3/5 × 12/13
= 20/65 + 36/65
= (20+36)/65
= 56/65

(ii) cos (A + B)
We know that cos (A +B) = cos A cos B – sin A sin B
So,
cos (A + B) = cos A cos B – sin A sin B
= 3/5 × 5/13 – 4/5 × 12/13
= 15/65 – 48/65
= -33/65

(iii) sin (A – B)
We know that sin (A – B) = sin A cos B – cos A sin B
So,
sin (A – B) = sin A cos B – cos A sin B
= 4/5 × 5/13 – 3/5 × 12/13
= 20/65 – 36/65
= -16/65

(iv) cos (A – B)
We know that cos (A -B) = cos A cos B + sin A sin B
So,
cos (A -B) = cos A cos B + sin A sin B
= 3/5 × 5/13 + 4/5 × 12/13
= 15/65 + 48/65
= 63/65

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