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20230312_202609.jpg (141.5 KiB) Прегледано 1947 пъти

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IMG_20220215_205717.jpg (99.98 KiB) Прегледано 1942 пъти

[tex]\cos^{2}{\alpha}-\sin^{2}{\alpha}=\cos{2\alpha}[/tex]

[tex]\tg{240 ^\circ }=\tg{(180 ^\circ +60 ^\circ )}=\tg{60 ^\circ }=\sqrt{3}[/tex]

[tex]\tg{150 ^\circ }=\tg{(180 ^\circ -30 ^\circ )} =-\tg{30 ^\circ }=-\frac{\sqrt{3}}{3}[/tex]

[tex]\frac{1}{\cotg{105 ^\circ }} = \tg{105 ^\circ }=\tg{(180 ^\circ -75 ^\circ )}=-\tg{75 ^\circ }=-\frac{\sin{75^\circ }}{\cos{75^\circ }}[/tex]

[tex]\sin{75^\circ }=\sin{(90^\circ -15^\circ)}=\cos{15^\circ} \Rightarrow \sin^{2}{75^\circ}=\cos^{2}{15^\circ}[/tex]


[tex]\tg{240^\circ}\cdot \tg{150^\circ} -\frac{1}{\cotg^{2}{105^\circ}}\cdot \cos^{2}{75^\circ}-\sin^{2}{15^\circ}=\sqrt{3}\cdot \left(-\frac{\sqrt{3}}{3}\right)-\left(-\frac{\sin{75^\circ }}{\cos{75^\circ }}\right)^{2}\cdot \cos^{2}{75^\circ }-\sin^{2}{15^\circ}=[/tex]

[tex]=-1+\frac{\sin^{2}{75^\circ }}{\cos^{2}{75^\circ }}\cdot \cos^{2}{75^\circ }-\sin^{2}{15^\circ}=-1+\underbrace{\cos^{2}{15^\circ}-\sin^{2}{15^\circ}}_{\cos{(2\cdot 15^\circ)}}=-1+\cos{30 ^\circ }=-1+\frac{\sqrt{3}}{2}=\frac{\sqrt{3}-2}{2}[/tex]