In a `Delta ABC `, if A,B,C are in AP, then `(a)/(c) sin2C+(c)/(a)(sin2A)` is equal to

Options

A. `(1)/(2)`
B. `(sqrt(3))/(2)`
C. 1
D. `sqrt(3)`

Answer

D

Explanation

Since ,`A,B and C ` are in AP
`therefore A+C=2B`
`rArr (A+C)+B=2B+B`
`rArr 180^@=3BrArr =60^@`....(ii)
`(a)/(c)(sin2C)+(c)/(a)(sin2A)`
`=(2a)/(c)(sinC cos C)+(2c)/(a)(sinA cos A)`
`=2a((sinC)/(c))cos C+2c((sinA)/(a))cosA`
`=2(ak)cos C +2(ck)cos A`
`=2(sin A cos C+sinB`
`[sin(a+phi)=sinthetacosphi+sinphicostheta]`
`=2 sin 60^@=2(sqrt(3)//2)=sqrt(3)`.