Sin^2 (arctg 3-arctg (1/2)) = 1/2

Sin^2 (arctg3-arctg (1/2)) = Sin (arctg3-arctg (1/2)) * Sin (arctg3-arctg (1/2)) =

= (Sin (arctg3) Cos (arctg (1/2)) - Cos (arctg3) Sin (arctg (1/2))) ²

Считаем по отдельности:

а) Sin (arctg3) = 3/√ (1 + 9) = 3√10/10

б) Сos (arctg (1/2)) = 1/√ (1 + 1/4) = 2√5/5

в) Сos (arctg3) = 1/√10 = √10/10

г) Sin (arctg (1/2) = 1/2/√5/4 = √5/5

Теперь наш пример:

(3√10/10 * 2√5/5 - √10/10 * √5/5) ² = (6√50/50 - √50/50) ² = (5√50/50) ²=

= (√50/10) ² = 50/100 = 0,5