We know that, A is in second quadrant, B is in first quadrant.
In the second quadrant, sine function is positive. cosine and tan functions are negative.
In first quadrant, all functions are positive.
By using the formulas,
cos A = – √(1 – sin2 A) and sin B = √(1 – cos2 B)
So let us find the value of cos A and sin B
cos A = – √(1 – sin2 A)
= – √(1 – (1/2)2)
= – √(1- 1/4)
= – √((4-1)/4)
= – √(3/4)
= -√3/2
sin B = √(1 – cos2 B)
= √(1-(√3/2)2)
= √(1- 3/4)
= √((4-3)/4)
= √(1/4)
= 1/2
We know, tan A = sin A / cos A and tan B = sin B / cos B
tan A = (1/2)/( -√3/2) = -1/√3 and
tan B = (1/2)/(√3/2) = 1/√3
(i) tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
= (-1/√3 + 1/√3) / (1 – (-1/√3) × 1/√3)
= 0 / (1 + 1/3)
= 0
(ii) tan (A – B) = (tan A – tan B) / (1 + tan A tan B)
= ((-1/√3) – (1/√3)) / (1 + (-1/√3) × (1/√3))
= ((-2/√3) / (1 – 1/3)
= ((-2/√3) / (3-1)/3)
= ((-2/√3) / 2/3)
= – √3