Question

if xcosA+ysinA=m and xsinA-ycosA=n, then prove that x^2+y^2=m^2+n^2​

Answer 1

Step-by-step explanation:

Given :

To prove if, x Cos A + y Sin A = m & x Sin A - y Cos A = n,

Then, x² + y² = m² + n²

Proof :

x Cos A + y Sin A = m  ...(i)

x Sin A - y Cos A = n   ...(ii)

By squaring and adding both the equations,

We get,

(x Cos A + y Sin A)² + (x Sin A - y Cos A)² = m² + n²

( x² Cos² A + y²Sin² A + 2xy Sin A Cos A) + (x² Sin² A + y² Cos² A - 2xy Sin A Cos A) = m² + n²

⇒ x² (Cos² A + Sin² A) + y² (Sin² A + y² Cos² A) = m² + n²

As,  Sin² A + Cos² A = 1,

⇒ x²(1) + y²(1) = m² + n²

⇒ x² + y² = m² + n²

Hence, Proved

Answer 2

$2x ^{2} = m {}^{2} + n {}^{2}$