Now, let's rewrite the expression in terms of sine and cosine:
(cos^2 t - ctg^2)/(sin^2 t - th^2 t = (cos^2 t - (cos^2 t/sin^2 t))/(sin^2 t - (sin^2 t/cos^2 t) = (cos^2 t - cos^2 t/sin^2 t)/(sin^2 t - sin^2 t/cos^2 t = [(cos^2 t sin^2 t - cos^2 t)/(sin^2 t)] / [(sin^2 t cos^2 t - sin^2 t)/(cos^2 t) = [(cos^2 t sin^2 t - cos^2 t)/sin^2 t] / [(sin^2 t cos^2 t - sin^2 t)/cos^2 t = [(cos^2 t(sin^2 t - 1))/sin^2 t] / [(sin^2 t(cos^2 t - 1))/cos^2 t = [(-cos^2 t)/sin^2 t] / [(sin^2 t)/cos^2 t = -(cos^2 t/sin^2 t) (cos^2 t/sin^2 t = -cot^2(t) cot^2(t = -cot^4(t)
Therefore, the simplified expression is -cot^4(t).
To simplify the expression (cos^2 t - ctg^2)/(sin^2 t - th^2 t), we will use trigonometric identities.
Recall that cotangent (ctg) and tangent (th) are reciprocal functions and can be expressed in terms of cosine and sine, respectively:
ctg(t) = 1/tan(t) = cos(t)/sin(t
th(t) = 1/cot(t) = sin(t)/cos(t)
Now, let's rewrite the expression in terms of sine and cosine:
(cos^2 t - ctg^2)/(sin^2 t - th^2 t
= (cos^2 t - (cos^2 t/sin^2 t))/(sin^2 t - (sin^2 t/cos^2 t)
= (cos^2 t - cos^2 t/sin^2 t)/(sin^2 t - sin^2 t/cos^2 t
= [(cos^2 t sin^2 t - cos^2 t)/(sin^2 t)] / [(sin^2 t cos^2 t - sin^2 t)/(cos^2 t)
= [(cos^2 t sin^2 t - cos^2 t)/sin^2 t] / [(sin^2 t cos^2 t - sin^2 t)/cos^2 t
= [(cos^2 t(sin^2 t - 1))/sin^2 t] / [(sin^2 t(cos^2 t - 1))/cos^2 t
= [(-cos^2 t)/sin^2 t] / [(sin^2 t)/cos^2 t
= -(cos^2 t/sin^2 t) (cos^2 t/sin^2 t
= -cot^2(t) cot^2(t
= -cot^4(t)
Therefore, the simplified expression is -cot^4(t).