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Let f(x)=sin^(2)(x +alpha)+sin^(2)(x +be...

Let `f(x)=sin^(2)(x +alpha)+sin^(2)(x +beta)-2cos(alpha-beta)sin(x+alpha)sin(x +beta)`. Which of the following is TRUE ?

A

f(x) is strictly increasing in `x in (alpha, beta)`

B

f(x) is strictly decreasing in `x in (alpha, beta)`

C

f(x) is strictly increasing in `x in (alpha,(alpha+beta)/(2))` and strictly decreasing in `x in ((alpha+beta)/(2),beta)`

D

f(x) is a constant function

Text Solution

Verified by Experts

The correct Answer is:
D
`f(x)=1-cos^(2)(x+alpha)+sin^(2)(x + beta) -2 cos (alpha-beta)sin (x + alpha) sin (x + beta)`
`=1-cos (2x + alpha + beta) cos (alpha - beta) - 2 cos (alpha - beta) sin (x + alpha) sin (x + beta)`
`= 1-cos (alpha - beta)[cos (2x + alpha + beta) - cos (2x+alpha+beta)+cos(alpha-beta)]`
`= sin^(2) (alpha-beta)`
which is a constant function.
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