if cosA/cosB = n, sinA/sinB = m, show that : (m^2 - n^2)sin^2 B = 1-n^2

We are given the equations:

cosA / cosB = n
sinA / sinB = m

We need to prove that:

(m² - n²) sin² B = 1 - n²

Step 1: Expressing sinA and cosA in terms of B
From the given equations:

cosA = n cosB
sinA = m sinB

Since we know that:

sin²A + cos²A = 1

Substituting the values of sinA and cosA:

(m sinB)² + (n cosB)² = 1

Step 2: Expanding the Equation
Expanding the squares:

m² sin² B + n² cos² B = 1

Rearrange the terms:

m² sin² B = 1 - n² cos² B

Using the identity cos² B = 1 - sin² B, substitute:

m² sin² B = 1 - n²(1 - sin² B)

Expanding:

m² sin² B = 1 - n² + n² sin² B

Step 3: Isolating the Required Expression
Rearrange the equation:

m² sin² B - n² sin² B = 1 - n²

Factor sin² B from the left-hand side:

(m² - n²) sin² B = 1 - n²

Conclusion:
Thus, we have successfully shown that:

(m² - n²) sin² B = 1 - n²