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Let f(x)={(sinx+cosx",",0 lt x lt (pi)/(...

Let `f(x)={(sinx+cosx",",0 lt x lt (pi)/(2)),(a",",x=pi//2),(tan^(2)x+"cosec"x",",pi//2 lt x lt pi):}`
Then its odd extension is

A

`{(-tan^(2)x-"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(-sinx+cosx",",-(pi)/(2) lt x lt 0 ):}`

B

`{(-tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(sinx-cosx",",-(pi)/(2) lt x lt 0 ):}`

C

`{(-tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(a",",x=-(pi)/(2)),(sinx-cosx",",-(pi)/(2) lt x lt 0 ):}`

D

`{(tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(sinx+cosx",",-(pi)/(2) lt x lt 0 ):}`

Text Solution

Verified by Experts

The correct Answer is:
B
For odd function,
`f(x)= -f(-x)`
` = -{(sin(-x)+cos(-x)",",0 lt -x lt pi//2),(a",",-x=pi//2),(tan^(2)(-x)+"cosec"(-x)",",pi//2 lt -x lt pi):}`
` ={(sinx-cosx",", -pi//2 lt x lt 0),(-a",",x= -pi//2),(-tan^(2)x+"cosec"x",",-pi lt x lt -pi//2 ):}`
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