Question

If tan x = 1, and 0 < x < 90, find cos(x/2)

Answer 1

Recall that

`\cos^2\frac x2=\frac{1+\cos x}2`

With
`0^\circ<x<90^\circ`, we have
`0^\circ<\frac x2<45^\circ`, so
`\cos\frac x2>0` and

`\cos\frac x2=\sqrt{\frac{1+\cos x}2}`

Also recall that

`\sec^2x=1+\tan^2x\implies\sec^2x=2\implies\sec x=\sqrt2\implies\cos x=\frac1{\sqrt2}`

Then

`\cos\frac x2=\sqrt{\frac{1+\frac1{\sqrt2}}2}=\frac{√(2+\sqrt2)}2`