Question

The number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1 is

(a) 0

(b) 2

(c) 4

(d) 6

Answer 1

sin x + sin y = 2 sin [(x+y)/2]cos[(x-y)/2]

sin(x+y) = 2 sin [(x+y)/2]cos[(x+y)/2]

cos[(x-y)/2] = cos[(x+y)/2]

cos[(x-y)/2] - cos[(x+y)/2] =0

2sin(x/2)sin(y/2) = 0 [we converted difference of cosines into a product]

x=0 or y=0.

So, there are four answers (0,1), (0,-1), (1,0), (-1,0).

Answer 2

Answer:

we know sin(x+y)=sinxcosy+cosxsiny

But it's given sin(x+y)=sinx+siny

∴cosx=cosy=1⟹x=y=2nπ±0

Also x,y must satisfy ∣x∣+∣y∣=1

Therefore there is no common pairs of (x,y)

which satisfies sin(x+y)=sinx+siny and ∣x∣+∣y∣=1

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