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Question: If \(x + y + z = 180^{o},\) then \(\cos 2x + \cos 2y - \cos 2z\) is equal to...

If x+y+z=180o,x + y + z = 180^{o}, then cos⁡2x+cos⁡2y−cos⁡2z\cos 2x + \cos 2y - \cos 2z is equal to

A

4sin⁡x.sin⁡y.sin⁡z4\sin x.\sin y.\sin z

B

1−4sin⁡x.sin⁡y.cos⁡z1 - 4\sin x.\sin y.\cos z

C

3cos⁡θ−4sin⁡θ3\cos\theta - 4\sin\theta

D

cos⁡A.cos⁡B.cos⁡C\cos A.\cos B.\cos C

Answer

1−4sin⁡x.sin⁡y.cos⁡z1 - 4\sin x.\sin y.\cos z

Explanation

Solution

cos⁡2x+cos⁡2y−cos⁡2z\cos 2x + \cos 2y - \cos 2z

=2cos⁡(x+y)cos⁡(x−y)−2cos⁡2z+1= 2\cos(x + y)\cos(x - y) - 2\cos^{2}z + 1

=2cos⁡(π−z)cos⁡(x−y)−2cos⁡2z+1= 2\cos(\pi - z)\cos(x - y) - 2\cos^{2}z + 1

=1−2cos⁡z{cos⁡(x−y)−cos⁡(x+y)}= 1 - 2\cos z\{\cos(x - y) - \cos(x + y)\}

=1−2cos⁡z2sin⁡xsin⁡y=1−4sin⁡xsin⁡ycos⁡z= 1 - 2\cos z2\sin x\sin y = 1 - 4\sin x\sin y\cos z.