класна работа

[doomaster]

опростете израза
cos(A+B)cos(A–B)+sin(A+B).sin(A-B)
4-sinA.sin(60°-A)cos(30°-A)

докажете тъждеството
4sinA.cosA.(1-tg²A)=sin4A
¯¯¯¯1+tg²A¯¯¯¯¯¯¯

cos2A =tg(45°-A)
¯¯¯¯1+sin2A¯¯¯
представете като произведения
3-4cos²A
1-2cosA+cos2A

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[tex]cos( \alpha + \beta )cos( \alpha - \beta )+sin( \alpha + \beta )sin( \alpha - \beta )=[/tex]
[tex]=(cos \alpha cos \beta- sin \alpha sin \beta)(cos \alpha cos \beta + sin \alpha sin \beta)+(sin \alpha cos \beta + cos \alpha sin \beta)(sin \alpha cos \beta - cos \alpha sin \beta)=[/tex]
[tex]=cos^2 \alpha cos ^2 \beta - sin^2 \alpha sin^2 \beta + sin^2 \alpha cos^2 \beta - cos^2\alpha sin ^2 \beta=[/tex]
[tex]=cos^2 \alpha (cos^2 \beta-sin^2 \beta)+sin ^2 \alpha (cos^2 \beta- sin^2 \beta)=[/tex]
[tex]=(cos^2 \beta- sin^2 \beta)(cos^2 \alpha+sin^2 \alpha)=[/tex]
[tex]=cos^2 \beta- sin^2 \beta=[/tex]
[tex]=cos 2 \beta[/tex]

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[tex]4-sin \alpha . sin (60 ^\circ - \alpha)cos(30^\circ - \alpha)=[/tex]
[tex]=4-sin \alpha (sin60^\circ cos \alpha - cos 60^ \circ sin \alpha)(cos 30^\circ cos \alpha + sin 30^\circ sin \alpha)=[/tex]
[tex]=4-sin \alpha (\frac{\sqrt{3}}{2} cos \alpha - \frac12 sin \alpha)(\frac{\sqrt{3}}{2} cos \alpha + \frac12 sin \alpha)=[/tex]
[tex]=4-sin \alpha (\frac34 cos ^2 \alpha - \frac14 sin^2 \alpha)[/tex]

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[tex]\frac{4 sin \alpha cos \alpha (1- tg ^2 \alpha)}{1+tg^2 \alpha}=sin 4 \alpha[/tex]
[tex]=\frac{2.2 sin \alpha cos \alpha (\frac{cos^2 \alpha - sin^2 \alpha}{cos ^2 \alpha})}{\frac{cos^2 \alpha + sin^2 \alpha}{cos ^2 \alpha}}=[/tex]
[tex]=\frac{2 sin 2 \alpha (\frac{cos 2 \alpha}{cos ^2 \alpha})}{\frac{1}{cos ^2 \alpha}}=[/tex]
[tex]=\frac{2 sin 2 \alpha cos 2 \alpha}{\cancel{cos^2 \alpha}}.\cancel{cos^2 \alpha}=[/tex]
[tex]=sin 4 \alpha[/tex]