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[r2d2]

Всички мерки на ъгли са в градуси!

tg 9 - tg 27 - tg 63 + tg 81 = ?

[Реклама]

[Relinquishmentor]

[tex]\tan 9 - \tan 27 - \tan 63 + \tan 81 = \frac{sin 9}{cos 9 } + \frac{sin 81}{cos 81} - \frac{sin 27}{cos 27} - \frac{sin 63}{cos 63} = [/tex]

[tex]\frac{sin 9 cos 81 + cos 9 sin 81}{cos 9 cos 81} - \frac{sin 27 cos 63 + cos 27 sin 63}{cos 27 cos 63} = \frac{sin (81 + 9)}{\frac{1}{2}(cos (9-81) + cos (81+9))} - \frac{sin (63+27)}{\frac{1}{2}(cos (27-63) + cos (63+27))} =[/tex]

[tex]2\frac{1}{cos 72} - 2\frac{1}{cos 36} = 2\frac{cos 36 - cos 72}{cos 36 cos 72} =[/tex]

[tex]2 \frac{-2sin{\frac{36 - 72}{2}}sin{\frac{36 + 72}{2}}}{sin (90 - 36)sin (90 - 72)} [/tex]

[tex]= 2.(-2)\frac{sin(-18)sin 54 }{sin 54 sin 18} = 4 [/tex]

[r2d2]

Това е ама може и малко по-просто!

[cub]

[tex] tg9-tg27-cotg27+cotg9=\frac{sin(9-27)}{ cos9cos27}+\frac{sin(27-9)}{sin9sin27 }=[/tex]

[tex]sin18.\frac{(-sin9sin27+cos9cos27)}{\frac{1}{ 4}sin2.9sin2.27 } =\frac{4.sin18cos36}{sin54sin18}[/tex]

[tex] \frac{4.cos36}{cos(90-54) }=4[/tex]