Question

The exact value of cos17pi/8 is:

Answer 1

`\cos\frac{17\pi}8=\cos\frac\pi8`

Recall the half-angle identity:

`\cos^2\frac\pi8=\frac{1+\cos\left(2\cdot\frac\pi8\right)}2\implies\cos\frac\pi8=\pm\sqrt{\frac{1+\cos\frac\pi4}2}`

`\frac\pi8` lies in the first quadrant, which means
`\cos\frac\pi8>0` so we should take the positive square root. Then

`\cos\frac\pi8=\sqrt{\frac{1+\frac1{\sqrt2}}2}=\frac{√(2+\sqrt2)}2`