Explanation:
tan(β+α) − (3+2√2) tan β = 0
Convert to sine and cosine:
sin(β+α) / cos(β+α) − (3+2√2) sin β / cos β = 0
sin(β+α) / cos(β+α) = (3+2√2) sin β / cos β
Cross multiply:
sin(β+α) cos β = (3+2√2) sin β cos(β+α)
sin(2β+α)
Rearrange:
sin(β+(β+α))
Angle sum formula:
sin β cos(β+α) + sin(β+α) cos β
Substitute:
sin β cos(β+α) + (3+2√2) sin β cos(β+α)
(4+2√2) sin β cos(β+α)
Rearrange:
(2+√2) (2 sin β cos(β+α))
Product to sum:
(2+√2) (sin(2β+α) + sin(-α))
Reflection:
(2+√2) (sin(2β+α) − sin α)
Since this equals sin(2β+α) from the beginning:
(2+√2) (sin(2β+α) − sin α) = sin(2β+α)
(2+√2) sin(2β+α) − (2+√2) sin α = sin(2β+α)
(1+√2) sin(2β+α) − (2+√2) sin α = 0
(1+√2) sin(2β+α) = (2+√2) sin α
sin(2β+α) = (2+√2) / (1+√2) sin α
Multiply by the conjugate:
sin(2β+α) = (2+√2)(1−√2) / ((1+√2)(1−√2)) sin α
sin(2β+α) = (2−2√2+√2−2) / (1−√2+√2−2) sin α
sin(2β+α) = (-√2) / (-1) sin α
sin(2β+α) = √2 sin α