Tg^2α+ctg^2α, если tgα+ctgα=4

Преобразуем выражение tg (α) + ctg (α):

tg (α) + ctg (α) = sin (α) / cos (α) + cos (α) / sin (α) = (sin^2 (α) + cos^2 (α)) / (sin (α) * cos (α)) = 1 / (sin (α) * cos (α)).

Так как

tg (α) + ctg (α) = 1 / (sin (α) * cos (α)) = 4,

то

sin (α) * cos (α) = 1/4.

Преобразуем выражение tg^2 (α) + ctg^2 (α):

tg^2 (α) + ctg^2 (α) = (1 - cos (2α)) / (1 + cos (2α)) + (1 + cos (2α)) / (1 - cos (2α)) = [ (1 - cos (2α)) ^2 + (1 + cos (2α)) ^2] / [ (1 + cos (2α)) * (1 - cos (2α)) ] = (1 - 2cos (2α) + cos^2 (2α) + 1 + 2cos (2α) + cos^2 (2α)) / (1 - cos^2 (2α)) = (2 + 2cos^2 (2α)) / sin^2 (2α) = 2 * (2 - sin^2 (2α)) / sin^2 (2α) = 2 * (2 - 2 * sin (α) * cos (α)) / (2 * sin (α) * cos (α)).

Учитывая, что

sin (α) * cos (α) = 1/4,

получим:

tg^2 (α) + ctg^2 (α) = 2 * (2 - 2 * 1/4) / (2 * 1/4) = 2 * (3/2) / (1/2) = 6.