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Question

If x+y+z=180°, then cos2x+cos2y-cos2z is equal to:


A

4sinxsinysinz

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B

1-4sinxsinycosz

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C

4sinxsinysinz-1

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D

cosAcosBcosC

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Solution

The correct option is B

1-4sinxsinycosz


Explanation for the correct option.

Step 1: Solve cos2x+cos2y.

cos2x+cos2y=2cos2x+2y2cos2x-2y2[cosA+cosB=2cosA+B2cosA-B2]cos2x+cos2y=2cosx+ycosx-y...1

Step 2: Solve cos2x+cos2y-cos2z

cos2x+cos2y-cos2z=2cosx+ycosx-y-2cos2z-1bycos2A=2cos2A-1=2cosx+ycosx-y-2cos2z+1=2cosx+ycosx-y-2cos2180°-x+y+1givenx+y+z=180°=2cosx+ycosx-y-2-cosx+y2+1bycos180°-θ=-cosθ=2cosx+ycosx-y-2cos2x+y+1=2cosx+ycosx-y-cosx+y+1=2cosx+y2sin2x2sin2y2+1[cosA-cosB=2sinA+B2sinB-A2]=4cosx+ysinxsiny+1=4cos180°-zsinxsiny+1=-4coszsinxsiny+1=1-4sinxsinycosz

Hence, option B is correct.


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