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# If tan (a+b-c) / tan (a-b+c) = tan c / tan bWhat is the value of sin 2a + sin 2b + sin 2c.

10 years ago

Dear karthik,

Multiply through by cos(a-b+c).cos(b).cos(a-b+c).cos(c) …

sin(a+b-c)cos(a-b+c).sin(b).cos(c) − sin(a-b+c).cos(a+b-c).sin(c).cos(b) = 0

½.sin(b).cos(c){ sin(2a) + sin(2b-2c) } − ½.sin(c).cos(b){ sin(2a) − sin(2b-2c) } = 0

½.sin(2a).sin(b-c) + ½.sin(2b-2c){ sin(b).cos(c) + sin(c).cos(b) } = 0

½.sin(2a).sin(b-c) + sin(b-c).cos(b-c).sin(b+c) = 0

½.sin(2a).sin(b-c) + ½.sin(b-c){ sin(2b) + sin(2c) } = 0

½.sin(b-c){ sin(2a) + sin(2b) + sin(2c) } = 0

∴ sin(2a) + sin(2b) + sin(2c) = 0

All the best.

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