Hi, need some help on the following question!! find the possible values of tan(1/2)x, if tanx=3/4 thanks A Ui need the method explained as clearly as possible pls

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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`tan(x)= 3/4`

``We need to find the value of `tan(1/2)x = tan(x/2)`

`We know that:`

`tan(x)= sin(x)/cos(x)`

`==> sin(x)/cos(x)= 3/4`

`==> 3cos(x)= 4sin(x)`

Now we know that `sin^2 x+ cos^2 x = 1 ==> cosx = sqrt(1-sin^2 x)`

`==> 3sqrt(1-sin^2 x)= 4sin(x)`

``Now we will square both sides:

`==> 9(1-sin^2 x) = 14sin^2 x`

`==> 9 - 9sin^2 x == 14sin^2 x`

`==> 25sin^2 x = 9`

`==> sin^2 x = 9/25`

`==> sin(x)= 3/5`

`==> cos(x)= 4/5` ..................(2)

Now we know that:

`cos(2x)= 1- 2sin^2 x`

`==> cos(x)= 1- 2sin^2 (x/2)`

`==> 4/5 = 1- 2sin^2 (x/2)`

`==> 2sin^2 (x/2) = 1- 4/5`

`==> 2sin^2 (x/2)= 1/5`

`==> sin^2 (x/2)= 1/10`

`==> sin(x/2)= 1/sqrt10`

``Now we know that:

`sin^2 (x/2) + cos^2 (x/2)= 1`

`==> cos(x/2)= sqrt(1-sin^2 (x/2))`

`==> cos(x/2)= sqrt(1- 1/10)`

`==> cos(x/2)= sqrt(9/10) = 3/sqrt10`

`==> tan(x/2)= sin(x/2) / cos(x/2)`

`==> tan(x/2)= (1/sqrt10) / (3/sqrt10) = 1/3`

`==> tan(x/2)= 1/3`

``

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