Question

How do you use the sum and difference formula to simplify `sec((17pi)/12)`?

Answer 1

`- csc (pi/12)`

Explanation:

First, simplify cos ((17pi)/12)
Use unit circle, property of complements and supplement arcs:
`cos ((17pi)/12) = cos ((5pi)/12 + pi) = - cos ((5pi)/12)`
`cos ((5pi)/12) = cos (-pi/12 + (6pi)/12) = cos (-pi/12 + pi/2) = sin (pi/2)`.
Finally,
`cos ((17pi)/12) = - cos ((5pi)/12) = - sin (pi/12)`
`sec ((17pi)/12) = - 1/sin (pi/12) = - csc (pi/12)`
Check by calculator:
`cos ((17pi)/12) = cos 255^@ = -0.259 --> sec = - 1/0.259 = -3.86`
`- sin (pi/12) = - sin 15^@ = - 0.259 -> - csc (pi/12) = - 3.86`. OK