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The sum of all x in [0,pi] which satisfy...

The sum of all `x in [0,pi]` which satisfy the equation sin `x+(1)/(2)cos x =sin^(2)(x+(pi)/(4))` is - (A) ` pi/6 ` (B) ` (5pi)/6 ` (C) ` pi ` (D) `2pi `

A

`(pi)/(4)`

B

`(5pi)/(6)`

C

`pi`

D

`2pi`

Text Solution

Verified by Experts

The correct Answer is:
C
`sin x +(1)/(2)cos x=sin^(2)(x+(pi)/(4))`
`sin x+(1)/(2)cos x =(1)/(2)(1-cos((pi)/(2)+2x))`
`sin x+(1)/(2)co x =(1)/(2)(1+sin 2x)`
`2 sin x +cos x = 1+sin x cos x`
`2 sin x cos x -2 sin x (1-cos x)=0`
`(1-cos x)-2 sin x(1-cos x)=0`
`(1-cos x)(1-2 sin x) = 0`
`1-cos x =0" "1-2sin x= 0`
`cos x =1" "sin x =(1)/(2)`
`x=0," "x=(pi)/(6),(5pi)/(6)`
sum `=0+(pi)/(6)+(5pi)/(6)=pi`
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