√3 * ctg5π/6 + tg π/12 ctg π/12 +2tg 7 π/4 +1

Предыдущий
вопрос Следующий
вопрос

To simplify the given expression, we first need to find the values of the trigonometric functions at the given angles.

Find the value of ctg(5π/6):
ctg(5π/6) = 1/tg(5π/6)
tg(5π/6) = tg(π - π/6) = -tg(π/6) = -√3
Therefore, ctg(5π/6) = 1/(-√3) = -1/√3 = -√3/3

Find the value of tg(π/12):
tg(π/12) = sin(π/12) / cos(π/12)

We know the values of sin(π/12) and cos(π/12) are not easily simplified and therefore we will leave it as tg(π/12).

Find the value of ctg(π/12):
ctg(π/12) = 1/tg(π/12)

Find the value of tg(7π/4):
tg(7π/4) = tg(π + 3π/4) = tg(3π/4) = -1

Now putting all the values back into the original expression:
√3 (-√3/3) + tg(π/12) 1/ tg(π/12) + 2 * (-1) + 1
= -3 + 1 + (-2) + 1
= -3 + 1 - 2 + 1
= -4

Therefore, the simplified value of the expression is -4.