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 = тАУ тИЪ(1 тАУ sin2 A) and sin B = тИЪ(1 тАУ cos2 B)
So let us find the value of cos A and sin B
cos A = тАУ тИЪ(1 тАУ sin2 A)
= тАУ тИЪ(1 тАУ (1/2)2)
= тАУ тИЪ(1- 1/4)
= тАУ тИЪ((4-1)/4)
= тАУ тИЪ(3/4)
= -тИЪ3/2
sin B = тИЪ(1 тАУ cos2 B)
= тИЪ(1-(тИЪ3/2)2)
= тИЪ(1- 3/4)
= тИЪ((4-3)/4)
= тИЪ(1/4)
= 1/2
We know, tan A = sin A / cos A and tan B = sin B / cos B
tan A = (1/2)/( -тИЪ3/2) = -1/тИЪ3 and
tan B = (1/2)/(тИЪ3/2) = 1/тИЪ3
(i) tan (A + B) = (tan A + tan B) / (1 тАУ tan A tan B)
= (-1/тИЪ3 + 1/тИЪ3) / (1 тАУ (-1/тИЪ3) ├Ч 1/тИЪ3)
= 0 / (1 + 1/3)
= 0
(ii) tan (A тАУ B) = (tan A тАУ tan B) / (1 + tan A tan B)
= ((-1/тИЪ3) тАУ (1/тИЪ3)) / (1 + (-1/тИЪ3) ├Ч (1/тИЪ3))
= ((-2/тИЪ3) / (1 тАУ 1/3)
= ((-2/тИЪ3) / (3-1)/3)
= ((-2/тИЪ3) / 2/3)
= тАУ тИЪ3