Question

Докажите тождество ctg2B-ctg4B=1/sin4B Плиз помогите

Answer 1

Ответ:

Тождество доказано.

Объяснение:

Доказать тождество

$ctg2\beta -ctg4\beta =\dfrac{1}{sin4\beta }$

Преобразуем левую часть. Для этого воспользуемся формулой

$ctg\alpha =\dfrac{cos\alpha }{sin\alpha }$

и приведем к общему знаменателю

$ctg2\beta -ctg4\beta =\dfrac{cos2\beta }{sin2\beta } -\dfrac{cos4\beta }{sin4\beta }=\dfrac{cos2\beta \cdot sin4\beta-cos4\beta \cdot sin2\beta }{sin2\beta \cdot sin4\beta }$

В числителе дроби воспользуемся формулой

$\sin\alpha \cdot \cos\beta -\cos\alpha \cdot \sin\beta =\sin (\alpha -\beta )$

и получим

$\dfrac{\sin (4\beta -2\beta )}{\sin2\beta \cdot\ sin4\beta } =\dfrac{\sin 2\beta }{\sin2\beta \cdot \sin4\beta } =\dfrac{1}{\sin4\beta }$

Левая часть равна правой части  и тождество доказано.

Answer 2

Solving the Trigonometric Identity

To prove the trigonometric identity ctg(2B) - ctg(4B) = 1/sin(4B), we can use the definition of cotangent and the fundamental trigonometric identities.

First, let's express cotangent in terms of sine and cosine: ctg(B) = cos(B) / sin(B)

Now, we can rewrite the left-hand side of the equation using the cotangent definition: ctg(2B) - ctg(4B) = (cos(2B) / sin(2B)) - (cos(4B) / sin(4B))

Next, we can use the double-angle and half-angle identities to simplify the expression.

Applying Trigonometric Identities

Using the double-angle identity: cos(2B) = 2cos(B)^2 - 1 sin(2B) = 2sin(B)cos(B)

We can substitute these expressions into the left-hand side of the equation.

Similarly, using the double-angle identity for cosine: cos(4B) = 2cos(2B)^2 - 1

We can substitute this expression into the left-hand side of the equation as well.

Simplifying the Expression

After substituting the double-angle identities, we can simplify the expression and manipulate it to match the right-hand side of the given identity: 1/sin(4B).

By simplifying the expression step by step, we can demonstrate that the left-hand side is equal to the right-hand side, thus proving the given trigonometric identity.

I hope this helps! Let me know if you'd like to see the step-by-step calculations.