The value of cos (a - b) has to be found given that sin a + sin b = 1 and cos a + cos b = 1/2.

cos (a - b) = cos a* cos b + sin a * sin b

sin a + sin b = 1

square both the sides

=> (sin a)^2 + (sin b)^2 + 2*sin a * sin b = 1 ...(1)

cos a + cos b = 1/2

square both the sides

(cos a)^2 + (cos b)^2 + 2*cos a* cos b = 1/4 ...(2)

(2) + (1)

=> (cos a)^2 + (cos b)^2 + 2*cos a* cos b + (sin a)^2 + (sin b)^2 + 2*sin a * sin b = 1 + 1/4

use the property (sin a)^2 + (cos a)^2 = 1

=> 2 + 2*cos a* cos b + 2*sin a * sin b = 5/4

=> 2*cos a* cos b + 2*sin a * sin b = 5/4 - 2

=> cos a* cos b + sin a * sin b = 5/8 - 1

=> cos (a - b) = -3/8

**The value of cos (a - b) = -3/8**

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