Question

If $acos\theta +bsin\theta =m$ and $asin\theta -bcos\theta =n$, then $a^2+b^2$ = ____

• A. $m^2-n^2$
• B. $m^2{n}^2$
• C. $n^2-m^2$
• D. $m^2+n^2$

Answer 1

The correct option is D $m^2+n^2$

Given
$acos\theta +bsin\theta =m$ and $asin\theta -bcos\theta =n$
$m^2+n^2=(asin\theta +bsin\theta {)}^2+(asin\theta -bsin\theta {)}^2$

Therefore,
$m^2+n^2=a^{2}co{s}^2\theta +b^{2}si{n}^2\theta +2abcos\theta sin\theta +a^{2}si{n}^2\theta +b^{2}co{s}^2\theta -2absin\theta cos\theta$
$m^2+n^2=(a^2+b^2)co{s}^2\theta +(a^2+b^2)si{n}^2\theta$
$m^2+n^2=(a^2+b^2)$