Question

The general solution of the equation $\sin x+\cos x=1$ is given by

• A. $x=2n\pi ,n\in I$
• B. $x=2n\pi +\frac{\pi }{2},n\in I$
• C. $x=n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4},n\in I$
• D. $x=\frac{n\pi }{2},n\in I$

Answer 1

Answer: (C)

$\sin x+\cos x=1$
$\Rightarrow \frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin \left(x+\frac{\pi }{4}\right)=\sin \frac{\pi }{4}$
$\Rightarrow x+\frac{\pi }{4}=n\pi +(-1{)}^n\frac{\pi }{4}$
$\Rightarrow x=n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4}$