The general solution of the trigonometrical equation sin⁡x+cos⁡x=1 for n=0,±1,… is given by

# The general solution of the trigonometrical equation  is given by

1. A

$x=2n\pi$

2. B

$x=2n\pi +\frac{\pi }{2}$

3. C

$x=n\pi +\left(-1{\right)}^{n}\frac{\pi }{4}-\frac{\pi }{4}$

4. D

none of these

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

### Solution:

We have

$\begin{array}{l}\mathrm{sin}x+\mathrm{cos}x=1\\ ⇒\sqrt{2}\left(\mathrm{sin}x\mathrm{cos}\frac{\pi }{4}+\mathrm{cos}x\mathrm{sin}\frac{\pi }{4}\right)=1\\ ⇒\mathrm{sin}\left(x+\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}\\ ⇒\mathrm{sin}\left(x+\frac{\pi }{4}\right)=\mathrm{sin}\frac{\pi }{4}\\ ⇒x+\frac{\pi }{4}=n\pi +\left(-1{\right)}^{n}\frac{\pi }{4},n\in Z\end{array}$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)