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Prove that (i) " cos " ((pi)/(4) + x...

Prove that
`(i) " cos " ((pi)/(4) + x) + " cos " ((pi)/(4)- x) =sqrt(2) " cos " x`
`(ii) " cos " ((3pi)/(4) + x) - "cos " ((3pi)/(4)-x) =- sqrt(2) " sin " x`

Text Solution

Verified by Experts

(i) LHS = cos `((pi)/(4) + x) + " cos " ((pi)/(4) -x)`
`=2 " cos " (pi)/(4) " cos " x [ :' " cos " (A+B) + " cos " (A-B) = 2 " cos " A " cos " B]`
`=(2 xx (1)/(sqrt(2)) " cos " x) = sqrt(2) " cos x " = RHS`
(ii) LHS `= " cos " ((3pi)/(4) + x) - " cos " ((3pi)/(4) -x)`
`=-2 " sin " .(3pi)/(4) " sin " x`
`[ :' " cos " (A+B) - " cos " (A-B) =- 2 " sin " A " sin "B]`
`=-2 "sin " (pi-(pi)/(4)) " sin " x`
`=-2 "sin " .(pi)/(2) " sin " x [ :' " sin " (pi-(pi)/(4)) = " sin " .(pi)/(4) ]`
`=(-2 xx (1)/(sqrt(2))) " sin x " =- sqrt(2) " sin " x = RHS`
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