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znanija.com/task/35269886

Упростить:   1 + tg3x - tg(60°+x )*tgx   * * * 1 + tg(3x) - tg(60+X) * tg(X) * * *

Решение :                  * * * tg(α+β) = (tgα +tgβ) / (1 - tgα*tgβ) * * *

tg3x =tg(x+2x)=(tgx+tg2x) / (1 -tgx*tg2x) =

( tgx +2tgx/(1 -tg²x) ) / (1 -tgx*2tgx/(1 -tg²x) = (3tgx - tg³x ) / (1-3tg²x)

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tg3x = tgx*(3 - tg²x ) / (1 - 3tg²x)  =

tgx*(√3 - tgx )(√3 + tgx )  /  (1 - √3tgx)(1 + √3tgx) =

tgx*(√3 - tgx )/ (1 + √3tgx) * (√3 + tgx ) / (1 - √3tgx) =

tgx* [ (tg60°- tgx )/ (1 + tg60°*tgx) ]* [ (tg60°+ tgx ) / (1 - tg60°*tgx) ]

tgx*tg(60°+ x) *tg(60°- x)    ⇒   tgx*tg(60°+ x) =  tg3x / tg(60°- x)

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1 + tg3x -  tg(60°+x )*tgx = 1 +tg3x - tg3x / tg(60°- x) =

1 +tg3x ( 1  - 1 / tg(60°- x) ) =  1 +tg3x (  1  - 1 / tg( 90° - (30°+x) ) =

 1 +tg3x (  1  - 1 / ctg(30°+x)  ) =  1 + tg3x ( 1 - tg(30°+x) )

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