First slide

Trigonometric Identities

Question

If $\sin θ + \cos θ = \sqrt{2} \cos θ$ then $\cos θ - \sin θ$ is equal to

Moderate

A

$\sqrt{2} \cos θ$
B

$\sqrt{2} \sin θ$
C

$\sqrt{2 (} \cos θ + \sin θ)$
D

none of these

Solution

We have,
$\begin{array}{l} \sin θ + \cos θ = \sqrt{2} \cos θ \\ \Rightarrow 1 + \sin 2 θ = 2 \cos^{2} θ \\ \Rightarrow 1 - \sin 2 θ = 2 - 2 \cos^{2} θ \\ \Rightarrow (\cos θ - \sin θ)^{2} = 2 \sin^{2} θ \\ \Rightarrow \cos θ - \sin θ = \sqrt{2} \sin θ \end{array}$

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