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Prove that (cos alpha+cos beta)^(2)+(sin...

Prove that `(cos alpha+cos beta)^(2)+(sin alpha -sin beta)^(2)=4 cos^(2) ((alpha+beta)/(2))`

A

`4cos^2((alpha-beta)/2)`

B

`4cos^2((alpha+beta)/2)`

C

`4cos^2(alpha-beta)`

D

`cos^2((alpha+beta)/2)`

Text Solution

Verified by Experts

The correct Answer is:
A
LHS=`(cos alpha+cos beta)^(2)+(sin alpha+sinbeta)^(2)`
`={2cos((alpha+beta)/(2))cos((alpha-beta)/(2))}^(2)+`
`{2sin((alpha+beta)/(2))cos((alpha-beta)/(2))}^(2)`
`=4 cos^(2)((alpha-beta)/(2)){cos^(2)(alpha+beta)/(2)+sin^(2)(alpha+beta)/(2)}`
`=4cos^(2)((alpha-beta)/(2))`=RHS.
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