Question

Find an exact value. Cos 75° negative square root of six plus square root of two quantity square root of six minus square root of two divided by four quantity negative square root of six plus square root of two divided by four square root of six minus square root of two

Answer 1

√2(√3 - 1)/4

Explanation:

To find an exact value for Cos75°, we use the compound angle formula. Since 75° = 45° + 30°, Cos75° = Cos(45° + 30°).

Using Cos(A + B) = CosACosB - SinASinB where A = 45° and B = 30°,

Cos75° = Cos(45° + 30°) = Cos45°Cos30° - Sin45°Sin30°

Now Cos45° = Sin45° = 1/√2 = √2/2, Cos30° = √3/2 and Sin30° = 1/2.

Substituting these values into the above equation, we have

Cos75° = Cos(45° + 30°)

= Cos45°Cos30° - Sin45°Sin30°

= √2/2 × √3/2 - √2/2 × 1/2

= √6/4 -√2/4

= √2(√3 - 1)/4