Question

If


`\cos(330°) = ( √(3) )/(2)`
Then prove that:

`\cos(165°) = - ( √(3) + 1)/(2 √(2) )`
Please help me!​

Answer 1

Answer: see proof below

Explanation:

Given: cos 330 =
`(\sqrt3)/(2)`

Use the Double-Angle Identity: cos 2A = 2 cos² A - 1

`\text{Scratchwork:}\quad \bigg((\sqrt3 + 2)/(2\sqrt2)\bigg)^2 = (2\sqrt3 + 4)/(8)`

Proof LHS → RHS:

LHS cos 165

Double-Angle: cos (2 · 165) = 2 cos² 165 - 1

⇒ cos 330 = 2 cos² 165 - 1

⇒ 2 cos² 165 = cos 330 + 1

Given:
`2 \cos^2 165 = (\sqrt3)/(2) + 1`

`\rightarrow 2 \cos^2 165 = (\sqrt3)/(2) + (2)/(2)`

Divide by 2:
`\cos^2 165 = (\sqrt3+2)/(4)`

`\rightarrow \cos^2 165 = \bigg((2)/(2)\bigg)(\sqrt3+2)/(4)`

`\rightarrow \cos^2 165 = (2\sqrt3+4)/(8)`

Square root:
`√(\cos^2 165) = \sqrt{(4+2\sqrt3)/(8)}`

Scratchwork:
`\cos^2 165 = \bigg((\sqrt3+1)/(2\sqrt2)\bigg)^2`

`\rightarrow \cos 165 = \pm (\sqrt3+1)/(2\sqrt2)`

Since cos 165 is in the 2nd Quadrant, the sign is NEGATIVE

`\rightarrow \cos 165 = - (\sqrt3+1)/(2\sqrt2)`

LHS = RHS
`\checkmark`