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if sin x + cos Y = A and cos x + sin y = b then tan of x-y/2 is equal to?

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2 years ago

```							Dear Student, tan(a/2) = sin(a/2)/cos(a/2) = 2sin(a/2)cos(a/2)/[2cos^2(a/2)] = sin(a)/(1+cos(a)) tan[(x-y)/2] = sin(x-y)/[1+cos(x-y)] ......(1) From (sinx+cosy)^2 + (cosx+siny)^2 = a^2 + b^2, we can get 2 + 2sinx cosy + 2cosx siny = a^2 + b^2 2[1+sin(x+y)] = (a^2+b^2) ......(2) From (sinx+cosy)^2 - (cosx+siny)^2 = a^2 - b^2, we can get 2sin(x-y) - cos2x + cos2y = a^2 - b^2 2sin(x-y) + 2sin(x+y)sin(x-y) = a^2 - b^2 2[1+sin(x+y)] sin(x-y) = a^2 - b^2 ......(3) Combine (2) and (3), and solve for sin(x-y), sin(x-y) = (a^2 - b^2)/(a^2 +b^2) ......(4) cos^2(x-y) = 1 - sin^2(x-y) = 1 - [(a^2 - b^2)/(a^2 +b^2)]^2 = 4a^2b^2/(a^2 + b^2)^2 So, cos(x-y) = 2ab/(a^2+b^2) ......(5) Substitute (4) and (5) into (1), tan[(x-y)/2] = sin(x-y)/[1+cos(x-y)] = [(a^2 - b^2)/(a^2 +b^2)]/[1+2ab/(a^2+b^2)] = (a^2-b^2)/(a+b)^2= (a-b)/(a+b)
```
2 years ago
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