Представете като произведение

[Гост]

1+sin a + cos a+ tg a

[S.B.]

1+sin a + cos a+ tg a

[tex]1 + sin \alpha + cos \alpha + tg \alpha =[/tex]

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

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

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

[tex]= 2 cos^{2} \frac{ \alpha }{2}( \frac{cos \alpha + sin \alpha }{cos \alpha } ) = 2 cos^{2} \frac{ \alpha }{2}( \frac{cos \alpha + cos(90 ^\circ- \alpha) }{cos \alpha } ) =[/tex]

[tex]= 2 cos^{2}\displaystyle \frac{ \alpha }{2}. \displaystyle \frac{1}{cos \alpha }.2cos\displaystyle \frac{ \alpha + 90 ^\circ - \alpha }{2}.cos\displaystyle \frac{ \alpha - 90 ^\circ + \alpha }{2} =[/tex]

[tex]= \displaystyle \frac{2 cos^{2}\displaystyle \frac{ \alpha }{2} }{cos \alpha }.2cos45 ^\circ .cos( \alpha -45 ^\circ ) = \displaystyle \frac{2 cos^{2}\displaystyle \frac{ \alpha }{2} }{cos \alpha } .2\displaystyle \frac{ \sqrt{2} }{2}.cos( \alpha - 45 ^\circ) =[/tex]

[tex]= 2 \sqrt{2}\displaystyle \frac{ cos^{2}\displaystyle \frac{ \alpha }{2}.cos(45 ^\circ - \alpha )}{cos \alpha }[/tex]