Double and half angles identities

Question

Find the exact value.

 

tan 7π/12= sin 7π/6 /2 / cos 7π/6 /2 

Answer 1

It will be negative since the angle is in the second quadrant.

 

cos 7π/6 = -√3 / 2

 

tan 7π/12 = -√[(1 - cos 7π/6)/(1 + cos 7π/6)]

 

= -√[(1  + √3 / 2)/(1 -  √3 / 2)]

 

= -√[(2 + √3)/(2 - √3)]

 

= -√{(2 + √3)²/[(2 - √3)(2 + √3)]}

 

= -√[(2 + √3)²]

 

= -2 - √3