Question

a)       (cos^2 t - ctg^2 t ) / (sin^2 t - tg^2 t) =?
б)       (tg t +1) / (1+ctg t) =?

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Answer 1

А) $frac{cos^{2}t-ctg^{2}t}{sin^{2}t-tg^{2}t}$

упростим числитель: $cos^{2}t-frac{cos^{2}t}{sin^{2}t}=frac{cos^{2}t*sin^{2}t-cos^{2}t}{sin^{2}t}=frac{cos^{2}t*(sin^{2}t-1)}{sin^{2}t}=-frac{cos^{2}t*(1-sin^{2}t)}{sin^{2}t}=$$-frac{cos^{4}t}{sin^{2}t}$

упростим знаменатель: $sin^{2}t-tg^{2}t=sin^{2}t-frac{sin^{2}t}{cos^{2}t}=frac{sin^{2}t*cos^{2}t-sin^{2}t}{cos^{2}t}=$$frac{sin^{2}t*(cos^{2}t-1)}{cos^{2}t}=-frac{sin^{2}t*(1-cos^{2}t)}{cos^{2}t}=-frac{sin^{4}t}{cos^{2}t}$

дробь: $frac{-frac{cos^{4}t}{sin^{2}t}}{-frac{sin^{4}t}{cos^{2}t}}= frac{cos^{4}t*cos^{2}t}{sin^{4}t*sin^{2}t}=$$frac{cos^{6}t}{sin^{6}t}=ctg^{6}t$

Б) $frac{tgt+1}{1+ctgt}= frac{tgt+1}{1+ frac{1}{tgt}}=frac{tgt+1}{frac{tgt+1}{tgt}}=frac{(tgt+1)*tgt}{tgt+1}=tgt$