Question

cosx+cosy+cosz=0=sinx+siny+sinz then tan(x-y)=?​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

tan(x-y) = ±$\sqrt{3}$

Step-by-step explanation:

$cosx+cosy+cosz=0\\=> cosx + cosy = -cosz\\=> cos^2x + cos^2y + 2.cosx.cosy = cos^2z...(i)\\\\\\sinx+siny+sinz=0\\=> sinx + siny = -sinz\\=> sin^2x + sin^2y + 2.sinx.siny = sin^2z...(ii)$

Summing up (i) and (ii) we get:

1 + 1 + 2(cosxcosy + sinxsiny) = 1

=> 2cos(x-y) = -1

=> cos(x-y) = -1/2

=> x-y is in either 2nd quadrant or 3rd quadrant

So, when x-y is in 2nd quadrant, then

$tan(x-y)\\= -\frac{\sqrt{(2)^{2} -(-1)^{2} } }{1} \\= -\frac{\sqrt{3} }{1}\\= -\sqrt{3}$

and when x-y is in 3rd quadrant, then

$tan(x-y)\\= \frac{\sqrt{(2)^{2} -(-1)^{2} } }{1} \\= \frac{\sqrt{3} }{1}\\= \sqrt{3}$