Given a triangle `DeltaABC` such that `sin^2 A + sin^2C = 1001.sin^2B`. Then the value of `(2(tanA+tanC)*tan^2B)/(tanA+tanB+tanC)` is

Options

A. `(1)/(2000)`
B. `(1)/(1000)`
C. `(1)/(500)`
D. `(1)/(250)`

Answer

D

Explanation

`sin^(2)A+sin^(2)C=1001sin^(2)B`
`rArr a^(2)+c^(2)=1001b^(2)` (using sine rule)
Now, `(2(tan A+ tan C).tan^(2)B)/(tan A + tan B + tan C)`
`=(2(tan A + tan C).tan^(2)B)/(tan A.tan B. tan C)`
`=2((cot A + cot C)/(cot B))`
`=(2(cos A sin C + sin A cos C))/(sin A. sin C. cos B) sin B`
`=(2 sin (pi-B).sin B)/(sin A sin C cos B)`
`=(2 sin^(2)B)/(sin A sin C cos B)`
`=(2xx2b^(2))/(2ab.cos B)`
`=(2xx2b^(2))/(a^(2)+c^(2)-b^(2))=(2xx2b^(2))/(1000b^(2))=(1)/(250)`