Question

What is the exact value of

Answer 1

4th option

Explanation:

using the addition formula for cosine

cos(a - b) = cosacosb + sinasinb

note that 15° = (45 - 30)° , then

cos15°

= cos(45 - 30)°

= cos45°cos30° + sin45°sin30°

= (
`(√(2) )/(2)` ×
`(√(3) )/(2)` ) + (
`(√(2) )/(2)` ×
`(1)/(2)` )

=
`(√(6) )/(4)` +
`(√(2) )/(4)`

=
`(√(6)+√(2) )/(4)`

=
`(√(2)+√(6) )/(4)`

Answer 2

`\cos 15^(\circ)= (√(2)+√(6))/(4)`

Explanation:

Trig Identity:

`\cos (A-B)=\cos A \cos B + \sin A \sin B`

`\begin{aligned}\implies \cos 15^(\circ)=\cos (45^(\circ)-30^(\circ)) & =\cos 45^(\circ) \cos 30^(\circ) + \sin 45^(\circ) \sin 30^(\circ)\\\\& = (√(2))/(2) \cdot (√(3))/(2) + (√(2))/(2) \cdot (1)/(2)\\\\& = (√(2)√(3))/(2 \cdot 2)+(√(2) \cdot 1)/(2 \cdot 2)\\\\& = (√(2\cdot 3))/(4)+(√(2))/(4)\\\\& = (√(6)+√(2))/(4)\\\\& = (√(2)+√(6))/(4)\end{aligned}`