Упр.341 Повторение ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) ctg(x)-sin(x)/(1-cos(x)); 3) (sin(a))^4-(sin(a))^2+(cos(a))^2;
2) sin(?)/(1+cos(?))+(1-cos(?))/sin(?); 4) tg(a)/(tg(a)+ctg(a)).

1. $$\ctg x-\frac{\sin x}{1-\cos x}=\frac{\cos x}{\sin x}-\frac{\sin x}{1-\cos x}$$
   $$=\frac{\cos x(1-\cos x)-\sin^2 x}{\sin x(1-\cos x)}$$
   $$=\frac{\cos x-\cos^2 x-\sin^2 x}{\sin x(1-\cos x)}$$
   $$=\frac{\cos x-1}{\sin x(1-\cos x)}=-\frac{1}{\sin x}.$$
2. $$\frac{\sin \varphi}{1+\cos \varphi}+\frac{1-\cos \varphi}{\sin \varphi}$$
   $$=\frac{\sin^2 \varphi+(1-\cos \varphi)^2}{\sin \varphi(1+\cos \varphi)}$$
   $$=\frac{\sin^2 \varphi+1-2\cos \varphi+\cos^2 \varphi}{\sin \varphi(1+\cos \varphi)}$$
   $$=\frac{2(1-\cos \varphi)}{\sin \varphi}=\frac{4\sin^2 \frac{\varphi}{2}}{2\sin \frac{\varphi}{2}\cos \frac{\varphi}{2}}=2\tg \frac{\varphi}{2}.$$
3. $$\sin^4 a-\sin^2 a+\cos^2 a=\sin^2 a(\sin^2 a-1)+\cos^2 a$$
   $$=\sin^2 a(-\cos^2 a)+\cos^2 a=\cos^2 a(1-\sin^2 a)=\cos^4 a.$$
4. $$\frac{\tg a}{\tg a+\ctg a}=\frac{\tg^2 a}{\tg^2 a+1}$$
   $$=\frac{\sin^2 a}{\cos^2 a}:\frac{1}{\cos^2 a}=\sin^2 a.$$

Ответ

1) $$-\frac{1}{\sin x}$$; 2) $$2\tg \frac{\varphi}{2}$$; 3) $$\cos^4 a$$; 4) $$\sin^2 a$$.