Упр.1298 ГДЗ Алимов 10-11 класс (Алгебра)

$$\frac{\tg \alpha+\tg \beta}{\ctg \alpha+\ctg \beta}=\frac{\tg \alpha+\tg \beta}{\frac{1}{\tg \alpha}+\frac{1}{\tg \beta}}=\frac{\tg \alpha+\tg \beta}{\frac{\tg \alpha+\tg \beta}{\tg \alpha \tg \beta}}=\tg \alpha \tg \beta.$$

$$(\sin \alpha+\cos \alpha)^2+(\sin \alpha-\cos \alpha)^2=$$

$$=\sin^2 \alpha+2\sin \alpha \cos \alpha+\cos^2 \alpha+\sin^2 \alpha-2\sin \alpha \cos \alpha+\cos^2 \alpha$$

$$=2\sin^2 \alpha+2\cos^2 \alpha=2(\sin^2 \alpha+\cos^2 \alpha)=2.$$

$$\frac{\sin\left(\frac{\pi}{4}+\alpha\right)+\cos\left(\frac{\pi}{4}+\alpha\right)} {\sin\left(\frac{\pi}{4}+\alpha\right)-\cos\left(\frac{\pi}{4}+\alpha\right)}$$

$$=\frac{\frac{\sqrt{2}}{2}\cos \alpha+\frac{\sqrt{2}}{2}\sin \alpha-\frac{\sqrt{2}}{2}\cos \alpha+\frac{\sqrt{2}}{2}\sin \alpha} {\frac{\sqrt{2}}{2}\cos \alpha+\frac{\sqrt{2}}{2}\sin \alpha+\frac{\sqrt{2}}{2}\cos \alpha-\frac{\sqrt{2}}{2}\sin \alpha} =\frac{\sqrt{2}\sin \alpha}{\sqrt{2}\cos \alpha}=\tg \alpha.$$

$$\frac{\sin \alpha+2\sin\left(\frac{\pi}{3}-\alpha\right)} {2\cos\left(\frac{\pi}{6}-\alpha\right)-\sqrt{3}\cos \alpha}$$

$$=\frac{\sin \alpha+2\left(\sin\frac{\pi}{3}\cos \alpha-\cos\frac{\pi}{3}\sin \alpha\right)} {2\left(\cos\frac{\pi}{6}\cos \alpha+\sin\frac{\pi}{6}\sin \alpha\right)-\sqrt{3}\cos \alpha}$$

$$=\frac{\sin \alpha+\sqrt{3}\cos \alpha-\sin \alpha} {\sqrt{3}\cos \alpha+\sin \alpha-\sqrt{3}\cos \alpha} =\frac{\sqrt{3}\cos \alpha}{\sin \alpha} =\sqrt{3}\ctg \alpha.$$