да се опростят изразите

[tedkoos]

Отново имам нужда от помощ.Ако някой може да покаже как се решават поне първите 2 примера ,ще му бъда много благодарен !
Прикачил съм ги ,за да е по-удобно.

[ammornil]

1) [tex]\frac{1+tg^{4}\alpha}{tg^{2}\alpha+cotg^{2}\alpha}[/tex]

използваме, че: [tex]cotg \alpha=\frac{1}{tg \alpha} \Longleftrightarrow cotg^{^{2}} \alpha=\frac{1}{tg^{^{2}} \alpha}[/tex]

[tex]\frac{1+tg^{^{4}}\alpha}{tg^{2}\alpha+\frac{1}{tg^{^{2}}\alpha}}=\frac{1+tg^{^{4}}\alpha}{\frac{tg^{^{4}}\alpha+1}{tg^{^{2}}\alpha}}=\frac{1}{\frac{1}{tg^{^{2}}\alpha}}=tg^{^{2}}\alpha[/tex]

[ammornil]

2) [tex]\frac{1-tg^{2}\alpha}{1+tg^{2}\alpha}+sin^{2}\alpha=\frac{1-\frac{sin^{2}\alpha}{cos^{2}\alpha}}{1+\frac{sin^{2}\alpha}{cos^{2}\alpha}}+sin^{2}\alpha=\frac{\frac{cos^{2}\alpha-sin^{2}\alpha}{cos^{2}\alpha}}{\frac{cos^{2}\alpha+sin^{2}\alpha}{cos^{2}\alpha}}+sin^{2}\alpha=\frac{cos^{2}\alpha-sin^{2}\alpha}{1}+sin^{2}\alpha=[/tex]

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

[ammornil]

3) [tex]\frac{sin^{2}\alpha-1}{1-cos^{2}\alpha}-\frac{1}{cos^{2}\alpha}+tg^{2}\alpha[/tex]

[tex]\frac{sin^{2}\alpha-1}{1-cos^{2}\alpha}-\frac{1}{cos^{2}\alpha}+tg^{2}\alpha=\frac{sin^{2}\alpha-1}{sin^{2}\alpha}-\frac{1}{cos^{2}\alpha}+\frac{sin^{2}\alpha}{cos^{2}\alpha}=\frac{cos^{2}\alpha.(sin^{2}\alpha-1)-sin^{2}\alpha+sin^{4}\alpha}{sin^{2}\alpha.cos^{2}\alpha}=[/tex]

[tex]=\frac{(1-sin^{2}\alpha).(sin^{2}\alpha-1)-sin^{2}\alpha+sin^{4}\alpha}{sin^{2}\alpha.cos^{2}\alpha}=\frac{sin^{2}\alpha-1-sin^{4}\alpha+sin^{2}\alpha-sin^{2}\alpha+sin^{4}\alpha}{sin^{2}\alpha.cos^{2}\alpha}=[/tex]

[tex]=\frac{sin^{2}\alpha-1}{sin^{2}\alpha.cos^{2}\alpha}=\frac{-(1-sin^{2}\alpha)}{sin^{2}\alpha.cos^{2}\alpha}=\frac{-cos^{2}\alpha}{sin^{2}\alpha.cos^{2}\alpha}=-\frac{1}{sin^{2}\alpha}[/tex]

[ammornil]

4) [tex]\frac{1}{sin\alpha}-cotg\alpha-\frac{sin\alpha+cos\alpha}{cos\alpha}+\frac{\frac{1}{cos\alpha}-1}{sin\alpha}=\frac{1}{sin\alpha}-\frac{cos\alpha}{sin\alpha}-\frac{sin\alpha+cos\alpha}{cos\alpha}+\frac{\frac{1-cos\alpha}{cos\alpha}}{sin\alpha}=[/tex]

[tex]=\frac{1-cos\alpha}{sin\alpha}-\frac{sin\alpha+cos\alpha}{cos\alpha}+\frac{1-cos\alpha}{sin\alpha.cos\alpha}=\frac{cos\alpha.(1-cos\alpha)-sin\alpha.(sin\alpha+cos\alpha)+1-cos\alpha}{sin\alpha.cos\alpha}=[/tex]

[tex]=\frac{cos\alpha-cos^{2}\alpha-sin^{2}\alpha-sin\alpha.cos\alpha+1-cos\alpha}{sin\alpha.cos\alpha}=-\frac{cos^{2}\alpha+sin^{2}\alpha+sin\alpha.cos\alpha-1}{sin\alpha.cos\alpha}=[/tex]

[tex]=-\frac{1+sin\alpha.cos\alpha-1}{sin\alpha.cos\alpha}=-\frac{sin\alpha.cos\alpha}{sin\alpha.cos\alpha}=-1[/tex]

[tedkoos]

Благодаря!