(cos^2 t - ctg^2)/(sin^2 t -th^2 t)

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вопрос

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).