((sint + cost)^2 - 1)\(ctg - sint cost) = 2 tg^2t

Предыдущий
вопрос Следующий
вопрос

To simplify the expression on the left side of the equation, we can start by expanding the numerator:

(sin(t) + cos(t))^2 - 1 = sin^2(t) + 2sin(t)cos(t) + cos^2(t) - 1
= sin^2(t) + cos^2(t) + 2sin(t)cos(t) - 1
= 1 + 2sin(t)cos(t) - 1
= 2sin(t)cos(t)

Next, we simplify the denominator:

ctg(t) - sin(t)cos(t) = cos(t)/sin(t) - sin(t)cos(t)
= cos^2(t)/(sin(t)cos(t)) - sin^2(t)/(sin(t)cos(t))
= (cos^2(t) - sin^2(t))/(sin(t)cos(t))
= cos(2t)/(sin(2t))
= cot(2t)

Therefore, the left side of the equation simplifies to:

(2sin(t)cos(t))/(cot(2t))
= 2(sintcost)/(cot(2t))
= 2(sintcost)/(ctg(2t))

Since cot(2t) = 1/tan(2t), we can simplify further:

2(sintcost)/(ctg(2t))
= 2(sintcost)/(1/tan(2t))
= 2(sintcost)*tan(2t)
= 2tg^2(t)

Therefore, the expression simplifies to 2tg^2(t).