Question

635. Упрости выражение е)

Answer 1

$\displaystyle \\\frac{\sin^2\alpha -\text{tg}^2\alpha }{\cos^2\alpha -\text{ctg}^2\alpha } =\frac{\sin^2\alpha -\dfrac{\sin^2\alpha }{\cos^2\alpha } }{\cos^2\alpha -\dfrac{\cos^2\alpha }{\sin^2\alpha } }=\frac{\dfrac{\cos^2\alpha\sin^2\alpha -\sin^2\alpha }{\cos^2\alpha } }{\dfrac{\sin^2\alpha\cos^2\alpha -cos^2\alpha }{\sin^2\alpha } } =\\\\\\=\frac{\dfrac{\sin^2\alpha (\cos^2\alpha -1)}{\cos^2\alpha } }{\dfrac{cos^2\alpha (\sin^2\alpha -1)}{sin^2\alpha } } =$

$=\displaystyle\frac{\dfrac{-\sin^2\alpha \sin^2\alpha }{\cos^2\alpha } }{\dfrac{-\cos^2\alpha\cos^2\alpha }{\sin^2\alpha } } =\frac{\dfrac{-\sin^4\alpha}{\cos^2\alpha } }{\dfrac{-\cos^4\alpha }{\sin^2\alpha } } =\frac{\dfrac{\sin^4\alpha}{\cos^2\alpha } }{\dfrac{\cos^4\alpha }{\sin^2\alpha } }=\frac{\sin^4\alpha }{\cos^2\alpha } \times\frac{\sin^2\alpha }{\cos^4\alpha } =\frac{\sin^6\alpha }{\cos^6\alpha } =\text{tg}^6\alpha$