Question

If sinx+cosy=a and cosx+siny=b,then tan (x-y)/2 is equal to a. a+b b.a-b c.a+b/a-b da-b/a+b

Answer 1

sin x + cos y = a
=> a = sin x + sin (90 - y) 
=> a = 2 sin [45 + (x-y)/2 ]  * Cos [45 - (x+y)/2 ]      --- (1)

sin y + cos x = b
=> b = sin y + sin (90 - x)
=> b = 2 sin [45 - (x-y)/2 ] * cos [ 45 - (x+y)/2 ]          ---- (2)

a - b = 2 cos [45 - (x+y)/2] * [ sin{45 +(x-y)/2} + sin{45-(x-y)/2} ]
        = 2 cos [45 - (x+y)/2] * 2 Sin {(x-y)/2}  * Cos 45     ... (3)

a + b = 2 Cos[45 - (x+y)/2] * 2 sin 45  Cos {(x-y)/2}         ...(4)

(a - b)/(a + b)    =  tan (x-y)/2