Question

Use a half-angle identity to find the exact value of cos15 .

a.
(sqrt2+sqrt3)/2
c.
(sqrt2-sqrt3)/2
b.
sqrt2+sqrt3
d.
sqrt2-sqrt3

Answer 1

Option a.
`(1)/(2) \sqrt{2+√(3) }`

Explanation:

In this question we have to find out the exact value of cos 15 by half angle identity.

As we know that
`cos((x)/(2) ) = \sqrt{(1+cosx)/(2) }` is the half angle identity.

From half this identity we can write cos15 as

`cos15 = cos(30)/(2)` =
`\sqrt{(1+cos30)/(2) }`

cos 15 =
`\sqrt{(1+(√(3) )/(2) )/(2) }`

=
`\sqrt{((√(3)+2)/(2) )/(2) } = \sqrt{((√(3)+2 )/(2))((1)/(2))}`

=
`\sqrt{(√(3)+2 )/(4) } = [tex](1)/(2) \sqrt{2+√(3) }`

So Option a.
`(1)/(2) \sqrt{2+√(3) }` is the right option.