Question

If (cos θ + sin θ) = $\sqrt{2}$ cos θ, then (cos θ − sin θ) = ?
(a) $\sqrt{2}$ sin θ
(b) $\sqrt{2}$ sec θ
(c) $\sqrt{2}$ cosec θ
(d) None of these

Answer 1

(a) $\sqrt{2}$ sin θ

$$\begin{gathered} \text{Given:}\left(\text{cos}\theta \text{+sin}\theta \right)=\sqrt{2}\text{cos}\theta \\ \text{We have:} \\ {\left(\text{cos}\theta \text{+sin}\theta \right)}^2+{(\text{cos}\theta -\text{sin}\theta )}^2=2\left({\text{cos}}^2\theta {\text{+sin}}^2\theta \right) \\ \text{=>}{\left(\sqrt{2}\text{cos}\theta \right)}^2+{(\text{cos}\theta -\text{sin}\theta )}^2=2\times 1 \\ \text{=>}{(\text{cos}\theta -\text{sin}\theta )}^2=2-2{\text{cos}}^2\theta \\ \text{=>}{(\text{cos}\theta -\text{sin}\theta )}^2=2(1-{\text{cos}}^2\theta ) \\ \text{=>}{(\text{cos}\theta -\text{sin}\theta )}^2=2{\text{sin}}^2\theta \\ \therefore (\text{cos}\theta -\text{sin}\theta )=\sqrt{2}\text{sin}\theta \end{gathered}$$