Question

If tan
$\alpha$
= 2xy/x² - y² prove that : sino = 2xy/x² + y²​

Answer 1

Given that :

1) 2 cos θ - sin θ = x    

2) cos θ - 3 sin θ = y

To Prove :

2x² + y² - 2xy = 5

Proof :  

2 x² = 2 ( 2cos θ - sin θ )²  =  2 + 6 cos² θ - 8 sin θ. cos θ    - (i)

y² = ( cos θ - 3 sin θ )² = 1 + sin² θ - 6 sin θ cos θ    - (ii)  

2 xy = 2 × (2cos θ - sin θ) (cos θ - 3 sin θ) = 4 + 2 sin² θ - 14 sin θ . cos θ   - (iii)

Now adding (i) , (ii) and (iii) we get :  

2x² + y² - 2xy = [2 + 6 cos² θ - 8 sin θ. cos θ] + [1 + sin² θ - 6 sin θ cos θ] - [4 + 2 sin² θ - 14 sin θ . cos θ]  

⇒ 2x² + y² - 2xy   = 6 cos² θ + 6 sin² θ - 1 = 6 (cos² θ + sin² θ) - 1 = 6 (1) - 1 = 5