SSC CGL 19 Oct 2014 Shift 2 Paper

3 sin θ + 5 cos θ = 5 --- (i)
5 sin θ – 3 cos θ = x (let)-- (ii)
On squaring and adding both the equations,
${\left(3\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }+5\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right)}^2$ + ${\left(5\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{–}3\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right)}^2$ = $5^2+{\mathit{x}}^2$
⇒ 9 $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ + 25 $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ + 30 sin θ .cos θ + 25 $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ + 9 $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ – 30 sin θ . cos θ = 25 + ${\mathit{x}}^2$
⇒ 9 $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ + 9 $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ + 25 $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ + 25 $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ = 25 + ${\mathit{x}}^2$
⇒ 9 ( $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ + $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ ) + 25 ( $\mathit{c}\mathit{o}{\mathit{s}}^2$ θ + $\mathit{s}\mathit{i}{\mathit{n}}^2$ θ ) = 25 + ${\mathit{x}}^2$
⇒ 9 + 25 = 25 + ${\mathit{x}}^2$
⇒ ${\mathit{x}}^2$ = 9 ⇒ x = ± 3