Question

What is Cos²A - Cos²B = ?

Answer 1

`=1/2(cos2A-cos2B)`

Explanation:

`cos^2A=1/2(1+cos2A)`
`cos^2B=1/2(1+cos2B)`

Thus,

`cos^2A-cos^2B=1/2(1+cos2A)-1/2(1+cos2B)`

`=1/2(1+cos2A-1-cos2B)`
`=1/2(cos2A-cos2B)`

Answer 2

-sin(A+B)sin(A-B)

Explanation:

`cos^2A-cos^2B=(cosA+cosB)(cosA-cosB)`----------`color(red)(1)`
Now,`cosA-cosB=2sin(A+B)//2 xx sin(B-A)//2`
Or, `cosA-cosB=-2sin(A+B)//2 xx sin(A-B)//2`

Also,`cosA+cosB=2cos(A+B)//2xxcos(A-B)//2`

Plugging in the values in `color(red)(1)`,and rearranging,we get,

`{-2cos(A+B)//2xxsin(A+B)//2} xx {2sin(A-B)//2 xx cos(A-B)//2}`
Thus,applying the formula `sin2x=2sinxcosx`,we have,
`cos^2A-cos^2B=-sin(A+B)sin(A-B)`