Упражнение тригонометрични функции

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[tex]\frac {1+cos a}{1-cos a} . tg^2 \frac {a}{2} - cos^2 a = sin ^2 a[/tex]
След прехвърляне, получаваме
[tex]\frac {\cancel {sin^2 \frac {a}{2}} + cos^2 \frac {a}{2}+cos^2 \frac {a}{2} \cancel {- sin^2 \frac {a}{2}}}{sin^2 \frac {a}{2} + cos ^2 \frac {a}{2} - (cos^2 \frac {a}{2} - sin ^2 \frac {a}{2})} . \frac {sin ^2 \frac {a}{2}}{cos ^2 \frac {a}{2}} = sin ^2 a + cos^2 a[/tex]
[tex]\frac { cos^2 \frac {a}{2}+cos^2 \frac {a}{2}}{sin^2 \frac {a}{2} \cancel {+ cos ^2 \frac {a}{2}} \cancel {- cos^2 \frac {a}{2}} + sin ^2 \frac {a}{2}} . \frac {sin ^2 \frac {a}{2}}{cos ^2 \frac {a}{2}} = 1[/tex]
[tex]\frac { 2 \cancel {cos^2 \frac {a}{2}}}{2 \cancel { sin ^2 \frac {a}{2}}} . \frac {\cancel {sin ^2 \frac {a}{2}}}{\cancel {cos ^2 \frac {a}{2}}} = 1[/tex]
[tex]\frac 22 = 1[/tex]
[tex]1=1[/tex]

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

[tex]cos a - 2 sin ^2 \frac {\pi - a}{2}=-1[/tex]
[tex]cos ^2 \frac {a}{2} - sin^2 \frac {a}{2} - 2 sin ^2 \frac {180 ^\circ - a}{2}=-1[/tex]
[tex]cos ^2 \frac {a}{2} - sin^2 \frac {a}{2} - 2 sin ^2 ( \frac {180 ^\circ }{2} - \frac { a}{2})=-1[/tex]
[tex]cos ^2 \frac {a}{2} - sin^2 \frac {a}{2} - 2 sin ^2 ( \frac {\cancel {2.}90 ^\circ }{\cancel {2}} - \frac { a}{2})=-1[/tex]
[tex]cos ^2 \frac {a}{2} - sin^2 \frac {a}{2} - 2 sin ^2 ( 90 ^\circ - \frac { a}{2})=-1[/tex]
[tex]cos ^2 \frac {a}{2} - sin^2 \frac {a}{2} - 2 cos ^2 \frac { a}{2}=-1[/tex]
[tex]- sin^2 \frac {a}{2} - cos ^2 \frac { a}{2}=-1[/tex]
[tex]- ( sin^2 \frac {a}{2} + cos ^2 \frac { a}{2})=-1[/tex]
[tex]-1 = -1[/tex]

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