Question

Using addition formula solve tan 15​

Answer 1

2 -
`√(3)`

Explanation:

Using the addition formula for tangent

tan(A - B) =
`(tanA-tanB)/(1+tanAtanB)` and the exact values

tan45° = 1 , tan60° =
`√(3)` , then

tan15° = tan(60 - 45)°

tan(60 - 45)°

=
`(tan60-tan45)/(1+tan60tan45)`

=
`(√(3)-1 )/(1+√(3) )`

Rationalise the denominator by multiplying numerator/ denominator by the conjugate of the denominator.

The conjugate of 1 +
`√(3)` is 1 -
`√(3)`

=
`((√(3)-1)(1-√(3)) )/((1+√(3))(1-√(3)) )` ← expand numerator/denominator using FOIL

=
`(√(3)-3-1+√(3) )/(1-3)`

=
`(-4+2√(3) )/(-2)`

=
`(-4)/(-2)` +
`(2√(3) )/(-2)`

= 2 -
`√(3)`