Question

Упростите:

$frac{1}{1-tg^2alpha}-frac{1}{1-ctg^2alpha}$

Правильный ответ $frac{1}{cos2alpha}$

Answer 1

$1-tg^2alpha = 1-frac{sin^2alpha}{cos^2alpha} = frac{cos2alpha}{cos^2alpha}</var><var>1-ctg^2alpha = 1-frac{cos^2alpha}{sin^2alpha} = - frac{cos2alpha}{sin^2alpha}$

тогда

 $frac{cos^2alpha}{cos2alpha} - (-frac{sin^2alpha}{cos2alpha}) = frac{1}{cos2alpha}$

Answer 2

1/(1-tg²α)-1/(1-ctg²α)=cos²α/(cos²α-sin²α)-sin²α/(sin²α-cos²α)=(cos²α+sin²α)/(cos²α-sin²α)=1/cos2α.