Question

If $acos\theta -bsin\theta =m$, and $asin\theta -bcos\theta =n$ then $a^2+b^2$ is equal to ______.

• 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 A $m^2-n^2$

$m^2-n^2=a^2\cdot co{s}^2\theta +b^2\cdot si{n}^2\theta -2ab\cdot sin\theta \cdot cos\theta -(a^2\cdot si{n}^2\theta +b^2\cdot co{s}^2\theta -2ab\cdot sin\theta \cdot cos\theta )$

$=a^2(co{s}^2\theta +si{n}^2\theta )+b^2(co{s}^2\theta +si{n}^2\theta )-2ab\cdot sin\theta \cdot cos\theta +2ab\cdot sin\theta \cdot cos\theta$

$=a^2+b^2$