Question

4 tg 15°/1 - tg^2 15°​

Answer 1

Ответ:

решение смотри на фотографии

Объяснение:

Answer 2

Let's calculate the expression:

4 * tan(15°) / (1 - tan^2(15°))

First, let's find the value of tan(15°):

tan(15°) = tan(45° - 30°)

We can use the trigonometric identity tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) * tan(B)):

tan(15°) = (tan(45°) - tan(30°)) / (1 + tan(45°) * tan(30°))

Now, we know that tan(45°) = 1 and tan(30°) = 1/√3:

tan(15°) = (1 - 1/√3) / (1 + 1/√3)

To rationalize the denominator, we can multiply both the numerator and denominator by √3:

tan(15°) = [(1 - 1/√3) * √3] / [(1 + 1/√3) * √3]

tan(15°) = [(√3 - 1) / √3] / [(√3 + 1) / √3]

Now, we can simplify the expression:

tan(15°) = (√3 - 1) / (√3 + 1)

Next, we'll square this value:

tan^2(15°) = [(√3 - 1) / (√3 + 1)]^2

To square the fraction, we square the numerator and denominator separately:

tan^2(15°) = [(√3 - 1)^2] / [(√3 + 1)^2]

Now, let's square the terms in the numerators:

(√3 - 1)^2 = 3 - 2√3 + 1 = 4 - 2√3 (√3 + 1)^2 = 3 + 2√3 + 1 = 4 + 2√3

So,

tan^2(15°) = (4 - 2√3) / (4 + 2√3)

Now, we can substitute these values back into the original expression:

4 * tan(15°) / (1 - tan^2(15°)) = 4 * [(√3 - 1) / (√3 + 1)] / [1 - (4 - 2√3) / (4 + 2√3)]

Now, let's simplify this expression further. First, simplify the denominator:

1 - (4 - 2√3) / (4 + 2√3) = 1 - (4 - 2√3) / (4 + 2√3) * (4 - 2√3) / (4 - 2√3)

1 - (4 - 2√3) / (4 + 2√3) * (4 - 2√3) / (4 - 2√3) = 1 - (16 - 8√3 - 8√3 + 12) / (16 - 12) = 1 - (28 - 16√3) / 4 = 1 - (7 - 4√3)

Now, let's simplify the numerator:

4 * (√3 - 1) = 4√3 - 4

So, the final expression is:

[4√3 - 4] / [1 - (7 - 4√3)] = (4√3 - 4) / (4√3 - 6)

Now, we can simplify it further by factoring out a 2 in the numerator and denominator:

(2√3 - 2) / (2√3 - 3)

So, 4 * tan(15°) / (1 - tan^2(15°)) simplifies to:

(2√3 - 2) / (2√3 - 3)