`(sqrt2-sinalpha-cosalpha)/(sinalpha-cosalpha)` is equal to

Options

A. `sec((alpha)/(2)-(pi)/(8))`
B. `cos((pi)/(8)-(alpha)/(2))`
C. `tan((alpha)/(2)-(pi)/(8))`
D. `cot((alpha)/(2)-(pi)/(2))`

Answer

C

Explanation

`(sqrt(2)-sin alpha-cosalpha)/(sin alpha-cos alpha)`
`=(sqrt(2)-sqrt(2)((1)/sqrt(2)sin alpha+(1)/sqrt(2)cos alpha))/(sqrt(2)((1)/sqrt(2)sin alpha-(1)/sqrt(2)cosalpha))`
`=(sqrt(2)-sqrt(2)cos(alpha-(pi)/(4)))/(sqrt(2)sin(alpha-(pi)/(4)))`
`=(sqrt(2)(1-cos theta))/(sqrt(2)sin theta)`.
Where `theta=a-(pi)/(4)=(2sin^(2)(theta//2))/(2sin(theta//2)cos(theta//2))`
`=tan((theta)/(2))=tan((alpha)/(2)-(pi)/(8))`.