Відповіді на вправи підручника: 125

125. Спростіть вираз:

1) 1 + sin² $\alpha$ + cos² $\alpha$ = 1 + 1 = 2;

2) 1 – sin² $\alpha$ + cos² $\alpha$ = cos² $\alpha$ + cos² $\alpha$ = 2cos² $\alpha$;

3) 1 + sin² $\alpha$ – cos² $\alpha$ = 1 – cos² $\alpha$ + sin² $\alpha$ = sin²$\alpha$ + sin²$\alpha$ = 2sin² $\alpha$;

4) –1 – sin² $\alpha$ + cos² $\alpha$ = –(sin² $\alpha$ + cos² $\alpha$) – sin² $\alpha$ + cos² $\alpha$ = –sin² $\alpha$ – cos² $\alpha$ – sin² $\alpha$ + cos² $\alpha$ = –2sin² $\alpha$;

5) 1 – sin² $\alpha$ = cos² $\alpha$;

6) 1 – cos² $\alpha$ = sin² $\alpha$;

7) –1 + sin² $\alpha$ = –(sin² $\alpha$ + cos² $\alpha$) + sin² $\alpha$ = –sin² $\alpha$ – cos² $\alpha$ + sin² $\alpha$ = –cos² $\alpha$;

8) –1 + cos² $\alpha$ = –(sin² $\alpha$ + cos² $\alpha$) + cos² $\alpha$ = –sin² $\alpha$ – cos² $\alpha$ + cos² $\alpha$ = –sin² $\alpha$;

9) $\frac{{1-\cos }^2\alpha }{{\sin }^2\alpha -1}$ = $\frac{{\sin }^2\alpha }{{-\cos }^2\alpha }$ = –tg² $\alpha$;

10) $\frac{{1-\sin }^2\alpha }{{\cos }^2\alpha -1}$ = $\frac{{\cos }^2\alpha }{{-\sin }^2\alpha }$ = –$\frac{1}{{\mathrm{tg}}^2\alpha }$.