If $x=4 \cos A+5 \sin A$ and $y=4 \sin A-5 \cos A$, then the value of $x^2+y^2$ is:

41

x = 4cosA + 5sinA

On squaring both side ,

x² = ( 4cosA + 5sinA )²

x² =  16cos²A + 25sin²A + 40sinA.cosA    ------(1)

And, y = 4sinA - 5cosA 

on squaring both side,

y² = ( 4sinA + 5cosA )²

y² =  16sin²A + 25cos²A - 40sinA.cosA     -------(2)

On adding equation 1 and 2.

x² + y² = 16 ( sin²A + cos²A ) + 25 ( sin²A + cos²A )

{ using, sin²A + cos²A = 1 }

x² + y²

= 16 + 25

= 41