Question

If x= a cos theta - b sin theta and y=a sin theta +b cos theta prove that x^2+y^2=a^2+b^2​

Answer 1

Answer:

$x = a \cos( \alpha ) - b \sin( \alpha ) \\ y = a \sin( \alpha ) + b \cos( \alpha )$

To prove:

${x}^{2} + {y}^{2} = {a}^{2} + {b}^{2}$

L.H.S.=>

$= > {x}^{2} + {y}^{2} \\ = > {(a \cos( \alpha ) - b \sin( \alpha ) ) }^{2} + {(a \sin( \alpha ) + b \cos( \alpha )) }^{2}$

$= > ( {a}^{2} { \cos}^{2} \alpha - 2ab \cos( \alpha ) \sin( \alpha ) + {b}^{2} {sin}^{2} \alpha ) + ( {a}^{2} {sin}^{2} \alpha + 2ab \sin( \alpha ) \cos( \alpha ) + {b}^{2} {cos}^{2} \alpha )$