Question
upstudy study bank question image url

46) \( \frac{\operatorname{sen} \alpha-\operatorname{sen} \beta}{\cos \alpha-\cos \beta}+\frac{\cos \alpha+\cos \beta}{\operatorname{sen} \alpha+\operatorname{sen} \beta}=0 \)

Ask by Grant Hodges. in Venezuela
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

La expresión se simplifica a \( 0 \), lo que demuestra la identidad.

Solution

1. Empezamos usando las fórmulas de transformación producto-suma: \[ \operatorname{sen} \alpha - \operatorname{sen} \beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right) \] \[ \cos \alpha - \cos \beta = -2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right) \] \[ \cos \alpha + \cos \beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right) \] \[ \operatorname{sen} \alpha + \operatorname{sen} \beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right) \] 2. Sustituimos en la primera fracción: \[ \frac{\operatorname{sen} \alpha-\operatorname{sen} \beta}{\cos \alpha-\cos \beta} = \frac{2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)}{-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)} = -\frac{\cos\left(\frac{\alpha+\beta}{2}\right)}{\sin\left(\frac{\alpha+\beta}{2}\right)} = -\cot\left(\frac{\alpha+\beta}{2}\right) \] 3. Sustituimos en la segunda fracción: \[ \frac{\cos \alpha+\cos \beta}{\operatorname{sen} \alpha+\operatorname{sen} \beta} = \frac{2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)}{2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)} = \frac{\cos\left(\frac{\alpha+\beta}{2}\right)}{\sin\left(\frac{\alpha+\beta}{2}\right)} = \cot\left(\frac{\alpha+\beta}{2}\right) \] 4. Sumamos ambas fracciones: \[ -\cot\left(\frac{\alpha+\beta}{2}\right) + \cot\left(\frac{\alpha+\beta}{2}\right) = 0 \] La expresión se simplifica a \( 0 \), lo que demuestra la identidad.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Bonus Knowledge

To solve the equation \[ \frac{\operatorname{sen} \alpha - \operatorname{sen} \beta}{\cos \alpha - \cos \beta} + \frac{\cos \alpha + \cos \beta}{\operatorname{sen} \alpha + \operatorname{sen} \beta} = 0, \] we can start by rearranging the first term and analyzing the structure. Notice that \[ \frac{\operatorname{sen} \alpha - \operatorname{sen} \beta}{\cos \alpha - \cos \beta} \] can be identified with the sine and cosine subtraction formulas. In addition, the second term can be simplified further with a common denominator. After performing some algebraic manipulation using identities such as \( \sin(a) - \sin(b) = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right) \) and \( \cos(a) - \cos(b) = -2 \sin\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right) \), the equation simplifies. Next, set each term equal to zero and examine when their numerators yield valid outputs. As we find specific angle values yielding true statements, we are effectively solving for \( \alpha \) and \( \beta \). Ultimately, exploring angle values and considering fundamental trigonometric properties can yield solutions such as \( \alpha = \beta + n\pi \) for integer \( n \) or unique intersection points for sine and cosine, which will simplify or validate your identity. Thus, the equation provides insights on practical use cases in geometry and physics, especially where angular relationships emerge!

Related Questions

Latest Trigonometry Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy