Question

Упростите тригонометрическое выражение, простым

Answer 1

$\frac{1}{ {sin}^{2} \alpha } \times \frac{1}{1 + {ctg}^{2} \alpha } = \\ = \frac{1}{ {sin}^{2} \alpha \times (tg \alpha ctg \alpha + {ctg}^{2} \alpha ) } = \\ = \frac{1}{ {sin}^{2} \alpha \times \frac{cos \alpha }{sin \alpha }( \frac{sin \alpha }{cos \alpha } + \frac{cos \alpha }{sin \alpha } ) } = \\ = \frac{1}{sin \alpha cos \alpha( \frac{sin \alpha }{cos \alpha } + \frac{cos \alpha }{sin \alpha } ) } = \\ = \frac{1}{ {sin}^{2} \alpha + {cos}^{2} \alpha} = \frac{1}{1} = 1$