identities..

alternate... though not a short one..

(sina + cosb)(sinb + cosa) = 2/9

=> (cosa.cosb + sina.sinb) + (sina.cosa + sinb.cosb) = 2/9

=> cos(a - b) + sin(a+b)cos(a-b)= 2/9

=> cos(a - b)(1 + sin(a+b)) = 2/9

squaring and adding the two gives,

sin(a + b)

now, put cos(a - b) here,

tan{(a-b)/2} = {1 - cos(a-b)}/sin(a - b)

i admitted its not a 'shorter' one :)