If cos α / cos β = m and cos α / sin β = n, then show that, (m² + n²) cos² β = n²?

To prove that \( (m² + n²) \cos² β = n² \) given the equations \( \frac{\cos α}{\cos β} = m \) and \( \frac{\cos α}{\sin β} = n \), we can start by expressing \( \cos α \) in terms of \( m \) and \( n \).

Step 1: Expressing cos α

From the first equation, we can write:

• \( \cos α = m \cos β \)

From the second equation, we have:

• \( \cos α = n \sin β \)

Step 2: Setting the equations equal

Since both expressions equal \( \cos α \), we can set them equal to each other:

• \( m \cos β = n \sin β \)

Step 3: Squaring both sides

Now, squaring both sides gives:

• \( m² \cos² β = n² \sin² β \)

Step 4: Using the Pythagorean identity

We know that \( \sin² β = 1 - \cos² β \). Substituting this into the equation yields:

• \( m² \cos² β = n² (1 - \cos² β) \)

Step 5: Rearranging the equation

Expanding the right side gives:

• \( m² \cos² β = n² - n² \cos² β \)

Now, rearranging terms results in:

• \( m² \cos² β + n² \cos² β = n² \)

Final Step: Factoring out cos² β

This can be factored as:

• \( (m² + n²) \cos² β = n² \)

This completes the proof, showing that \( (m² + n²) \cos² β = n² \) holds true.