If √2cosxcosy=1/2 and tanx+coty =2.find the value of (x-y).
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Answer:For this problem, you need the following identities
sin(2x) = 2 sinxcosx
sin(x+y) = sinxcosy + sinycosx
sinx + siny = 2sin[(x+y}/2]cos[(x-y)/2]
ab = (sinx + siny)(cosx + cosy)
= sinxcosx + (sinxcosy + sinycosx) + sinycosy
= ½sin(2x) + sin(x+y) + ½sin(2y)
= sin(x+y) + ½[sin(2x) + sin(2y)]
= sin(x+y) + sin(x+y)cos(x-y)
= sin(x+y) [1 + cos(x-y)] (1)
a2 = (sinx + siny)2 = sinx2 + 2sinxsiny + siny2
b2 = (cosx + cosy)2 = cosx2 + 2cosxcosy + cosy2
a2 + b2 = 2 + 2 cos(x-y)
= 2 [1 + cos(x-y)] (2)
From (1) and (2)
sin(x+y) = 2ab/(a2 + b2)
Step-by-step explanation:
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