Question

Cos^2t-ctg^2t/sin^2t-tg^2t

Answer 1

Скобки на месте ставить надо. Скорее всего вид задания таков:
(cos^2t-ctg^2t)/(sin^2t-tg^2t)
cos^2t-ctg^2t=cos^2t-cos^2t/sin^2t=(cos^2t*sin^2t-cos^2t)/sin^2t=
=(-cos^2t(1-sin^2t))/sin^2t=-cos^4t/sin^2t - числитель
sin^2t-tg^2t=sin^2t-sin^2t/cos^2t=(sin^2t*cos^2t-sin^2t)/cos^2t=
=(-sin^2t(1-cos^2t))/cos^2t=-sin^4t/cos^2t - знаменатель
Делим числитель на знаменатель
-cos^4t/sin^2t:(-sin^4t/cos^2t)=cos^6t/sin^6t=ctg^6t

Answer 2

(cos²t- cos²t/sin²t)/(sin²t - sin²t/cos²t)=cos²t(sin²t-1)/sin²t : sin²t(cos²t-1)/cos²t=
=-cos^4t/sin²t :-sin^4t/cos²t=cos^4t/sin²t * cos²t/sin^4t=cos^6t/sin^6t=ctg^6t

Answer 3

To simplify the expression (cos^2t - ctg^2t) / (sin^2t - tg^2t), we can use trigonometric identities to rewrite the terms in a more convenient form.

Let's start by expressing cotangent (ctg) and tangent (tg) in terms of sine (sin) and cosine (cos):

ctg(t) = 1 / tan(t) = cos(t) / sin(t) tg(t) = sin(t) / cos(t)

Substituting these identities into the original expression, we have:

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

Simplifying further, we can expand the squared terms:

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

To simplify the expression, we can find a common denominator for both terms:

= [(cos^2t * sin^2t) - cos^2t] / [(sin^2t * cos^2t) - sin^2t]

Factoring out a common factor in the numerator and denominator:

= [cos^2t * (sin^2t - 1)] / [sin^2t * (cos^2t - 1)]

Using the trigonometric identity sin^2t + cos^2t = 1, we can rewrite the expression further:

= [cos^2t * (-cos^2t)] / [(-sin^2t) * (-sin^2t)]

Simplifying the negative signs:

= [cos^2t * (-cos^2t)] / [sin^2t * sin^2t]

= - (cos^4t) / (sin^4t)

Therefore, the simplified expression is -cos^4t / sin^4t.