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Solution :

Let `tanA = 3k, tanB = 4k, tanC = 5k`<br>
As, `A+B+C = pi`<br>
`:. tanA+tanB+tanC = tanAtanBtanC`<br>
`=>3k+4k+5k = 3k*4k*5k`<br>
`=>12k = 60k^3`<br>
`=>k^2 = 1/5`<br>
`=> k = 1/sqrt5`<br>
`:. tan A = 3/sqrt5, tanB = 4/sqrt5, tanC = 5/sqrt5 = sqrt5`<br>
Now, `sinx = tanxcosx = tanx/secx = tanx/(sqrt(1+tan^2x))`<br>
`:. sin A = 3/sqrt14, sin B = 4/sqrt21, sinC = sqrt5/sqrt6`<br>
`:. sinAsinBsinC = (3/sqrt14)(4/sqrt21)(sqrt5/sqrt6) = (2sqrt5)/7`<br>
So, option `(b)` is the correct option.<br>Transcript

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00:00 - 00:59 | ABC if tan a tan B resistance is equal to 3 ratio 4 ratio 5 then the value of sin a into sin theta into sin C is equal to what so here is some options which is given angle have to tell that current 21 ok doing triangle ABC tan a tan B and C is equal to 35 and here we can say that are 10 is equal to 3 K10 P is equal to four k and N C is equal to 5 p.m. that in triangle abc Commission tan a is equals to this site so here we can |

01:00 - 01:59 | on which like this it will be seen that we harvest 12th ke which is equal to 60k IU and their finally we have a 5 case now ke is equal to 1 by root 5 which is ton is equal to 3 by 5 and 10 B will be equal to 5 and will be equal to 55 profile here we have seen so now we can do one thing more that is |

02:00 - 02:59 | equal to 3 by root 5 + 9 and fine be because 24 25 + 60 = 25 by 25 25 picture pic for 2014 and find we hold by root 21 and fine Si will be 5 ok now considered find a b t a into sin b into x minus 3 root |

03:00 - 03:59 | 4221 and route 30 where we finally have it 60 by route 88 2018 and now it will be trained by 725 which will be equal to 2 root 5 by 7 and this is our final you here we can check that which one is correct here we can see that 2 root 5 23557 we can see that b is correct option for this ok thank you |