=λ This can be written as cos‌x=λ‌cos(x−2y) Expanding the right side using the angle subtraction formula for cosine: cos‌x=λ[cos‌x‌cos‌2‌y+sin‌xsin‌2y] Rearranging:
cos‌x−λ‌cos‌x‌cos‌2‌y=λsin‌xsin‌2y Factoring out cos‌x on the left side: cos‌x(1−λ‌cos‌2‌y)=λsin‌xsin‌2y Dividing both sides by cos‌x‌s‌i‌n‌2y : ‌
1−λ‌cos‌2‌y
sin‌2y
=‌
λsin‌x
cos‌x
Simplifying using the double angle formulas: ‌‌
1−λ(1−2sin‌2y)
2sin‌y‌cos‌y
=λ‌tan‌x ‌‌
1−λ+2λsin‌2y
2sin‌y‌cos‌y
=λ‌tan‌x ‌‌
1−λ
2sin‌y‌cos‌y
+‌
2λsin‌2y
2sin‌y‌cos‌y
=λ‌tan‌x ‌‌
1−λ
2sin‌y‌cos‌y
+λ‌tan‌y=λ‌tan‌x Rearranging to isolate tan(x−y) : λ‌tan‌x−λ‌tan‌y=‌
1−λ
2sin‌y‌cos‌y
λ(tan‌x−tan‌y)=‌
1−λ
2sin‌y‌cos‌y
Using the tangent subtraction formula: λ‌tan(x−y)=‌