Question

Prove that the image of point `P(costheta, sin theta)` in the line having slope `tan(alpha//2)` and passing through origin is `Q(cos(alpha-theta),sin(alpha-theta))`.

Answer 1

Clearly, `OP=OQ=1`
Now, we have to prove that Q is the image of P in the line OR which has slope tan `(alpha//2)`.
Triangle `POQ` is isosceles triangle.
If Q is the image of P in line OR, then OR is the perpendicular bisector of PQ.
We have to prove that `angle QOM=alpha-theta`.
`angle ROQ=angle POR=theta-(alpha//2)`
`therefore angle QOM=angleROM-angle ROQ`
`=(alpha//2)-(theta-(alpha))=alpha-theta`