Point D,E are taken on the side BC of an acute angled triangle ABC,, such that `BD = DE = EC`. If `angle BAD = x, angle DAE = y and angle EAC = z` then the value of `(sin(x+y) sin (y+z))/(sinx sin z)` is ______

Answer

4

Explanation

In `DeltaABD`,
`(a)/(3 sin x) = (AD)/(sin B)` ...(1)
In `DeltaABE`,
`(2a)/(3 sin (x +y)) = (AE)/(sin B)`
Dividing Eq. (1) by Eq. (2), we...(2)
`(sin (x+y))/(2 sinx) = (AD)/(AE)` ...(3)
In `DeltaADC`,
`(2a)/(3 sin (y + z)) = (AD)/(sin C)` ...(4)
In `DeltaAEC`,
`(a)/(3 sin z) = (AE)/(sin C)` ...(5)
Dividing Eq. (5) by Eq. (4), we get
`(sin (y +z))/(2 sin z) = (AE)/(AD)`...(6)
Multiplying Eq. (3) and Eq. (6) , we get
`:. (sin (x +y) sin (y + z))/(sin x.sinz) = 4`