If sin(x +
y) = 1 and sin(x -
y) = (1/2), then what is the value of sinx cosx + 2sin2y + cos2y? 

We have given, sin(x + y) = 1 and sin(x - y) = 1/2 So, sin(x + y) = 1 Or, sin(x + y) = sin90° Or, (x + y) = 90° ......(i) Again, sin(x - y) = 1/2 Or, sin(x - y) = sin30° Or, (x - y) = 30° ......(ii) By adding (i) and (ii) we get, 2x = 120° Or, x = 60° Putting the value of 'x' in the eq (i) we get, y = 30° Now, sinx cosx + 2sin²y + cos²y = sin60° X cos60° + 2 X sin²30° + cos²30° = = (1/2) X (√3/2) + 2 X (1/2)² + (√3/2)² = (1/2) X (√3/2) + 2 X (1/4) + (3/4) = (√3/4) + (1/2) + (3/4) = {(√3 + 2 + 3)/4} = {(√3 + 5)/4}