What is the value of cos 2x in terms of m and n?

Question

Given that cos x + sin x = m and cos x - sin x = n, where n and m are constants, write down in terms of m and n, the value of cos 2x!

Answer 1

m=cosx+sinx

n = cosx-sinx

mn = (cosx+sinx)(cosx-sinx) = cos²(x)-sin²(x) = cos(2x) by the cosine double angle formula

cos(2x) = mn

Answer 2

WE square both equations: cos²x + 2cosxsinx + sin²x = m²; cos²x - 2cosxsinx + sin²x = n²; or

1 + sin2x = m²; 1 - sin2x = n²; 2sin2x = m² - n²; sin2x = (m² - n²)/2;

cos2x = ±√[1 - (m² - n²)²/4] = ±√[4 - (m² - n²)²]/2