find the exact value of cos 157.5°.

Cos 157.5º=-cos (180º-157.5º) = - cos 22.5=-cos (45/2)

cos Ф/2=⁺₋√ ((1+cosФ) / 2).

In this case 157.5º is in the second quadrant, therefore we use the following equation:

cos Ф/2=-√ ((1+cosФ) / 2). (we will have a negative number)

cos 157.5º=-cos (45/2) = - √ ((1+cos 45º) / 2)

=-√ ((1+√2/2) / 2)

=-√ ((2+√2) / 4)

=-√ (2+√2) / 2 (≈-0.92387 ...)

Answer: cos 157.5º = - √ (2+√2) / 2