4 задачи по тригонометрия!

[Гост]

1 . [tex]\frac{tg\alpha + tg\beta}{ tg(\alpha+\beta)} + \frac {tg\alpha - tg\beta} {tg(\alpha-\beta)} = 2[/tex]

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

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

4. [tex]\frac{4sin\alpha - 5cos\alpha}{3cos\alpha-sin\alpha}[/tex]

[mail_dinko]

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

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

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