Q112 / 180 · KCET 2013 · Mathematics

The general solution of $\sin x-\cos x=\sqrt{2}$, for any integer $n$ is

  • Correct answer
    A
    $2 n \pi+\frac{3 \pi}{4}$
  • B
    $n \pi$
  • C
    $(2 n+1) \pi$
  • D
    $2 n \pi$

Solution

$\begin{array}{ll}\text { Given, } & \sin x-\cos x=\sqrt{2} \\ \Rightarrow & \frac{1}{\sqrt{2}} \sin x-\frac{1}{\sqrt{2}} \cos x=1 \\ \Rightarrow & \sin x \cdot \sin \frac{\pi}{4}-\cos x \cdot \cos \frac{\pi}{4}=1 \\ \Rightarrow & -\cos \left(x+\frac{\pi}{4}\right)=+1 \\ \Rightarrow & \cos \left(x+\frac{\pi}{4}\right)=\cos \pi \\ \Rightarrow & x+\frac{\pi}{4}=2 n \pi+\pi \\ \therefore & x=2 n \pi+\frac{3 \pi}{4}\end{array}$

Hence, the correct answer is option: 'A'.

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