Question

Find the value exact value of tan 105 degrees. Use a half-angle identity.

Answer 1

There is no half-angle identity for the tangent function, so we cannot find the exact value of tan 105 degrees using a half-angle identity.

Step-by-step explanation:

To find the exact value of tan 105 degrees using a half-angle identity, we can use the identity: tan(x) = sin(x) / cos(x). However, there is no half-angle identity for the tangent function. Therefore, we cannot find the exact value of tan 105 degrees using a half-angle identity.

Answer 2

The exact value of
`tan(105\°)` is
`-2-√(3)`

Step-by-step explanation:

The half-angle identity for 'tangent' is.....

`tan((\theta)/(2))=(sin(\theta))/(1+cos(\theta))`

Here,
`(\theta)/(2)=105\° \Rightarrow \theta=210\°`

Plugging the value of
`\theta` into the above formula.....

`tan((210\°)/(2))=(sin(210\°))/(1+cos(210\°))\\ \\ tan(105\°)=(-(1)/(2))/(1+(-(√(3))/(2)))\\ \\ tan(105\°)=(-(1)/(2))/(1-(√(3))/(2))\\ \\ tan(105\°)=(-(1)/(2))/((2-√(3))/(2))\\ \\tan(105\°)=-(1)/(2-√(3))=-((2+√(3)))/((2-√(3))(2+√(3)))=-((2+√(3)))/(4-3)=-((2+√(3)))/(1)\\ \\ tan(105\°)=-2-√(3)`