Bihar Police Constable Model Paper 1

Given:
x = m sin θ – n cos θ …i)
y = m cos θ + n sin θ …ii)
Concept:
Basic trigonometric identities and algebra
Calculation:
Squaring equation (i) and (ii)
${\mathit{x}}^2$ = ${\mathit{m}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }$ + 2mn sin θ cos θ …iii)
${\mathit{y}}^2$ = ${\mathit{m}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }$ – 2mn sin θ cos θ …iv)
Adding equation (iii) and (iv)
⇒ ${\mathit{x}}^2+{\mathit{y}}^2$ = ${\mathit{m}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }$ + 2mn sin θ cos θ + ${\mathit{m}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }$ – 2mn sin θ cos θ
⇒ ${\mathit{x}}^2+{\mathit{y}}^2$ = ${\mathit{m}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }$ + ${\mathit{m}}^2\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }+{\mathit{n}}^2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }$
Applying the identity $\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{\theta }$ = 1
⇒ ${\mathit{x}}^2+{\mathit{y}}^2$ = ${\mathit{m}}^2+{\mathit{n}}^2$