Question

Solve the equation
$\left(i\right)\sin x+\cos x=\sqrt{2}$

Answer 1

$\sin x+\cos x=\sqrt{2}$

$\Rightarrow \frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=1$

$\Rightarrow \sin \frac{\pi }{4}\sin x+\cos \frac{\pi }{4}\cos x=1$

$\Rightarrow \cos (x-\frac{\pi }{4})=1$

$[\because \cos (A-B)=\cos A\cos B+\sin A\sin B]$

$\Rightarrow \cos (x-\frac{\pi }{4})=\cos 0$

We know the general solution of $\cos x=\cos \alpha$ is $x=2n\pi \pm \alpha ,n\in Z$

So,
$(x-\frac{\pi }{4})=2n\pi \pm 0$

$\Rightarrow x=2n\pi +\frac{\pi }{4}$

$\Rightarrow x=(8n+1)\frac{\pi }{4}$

Hence, the general solution is $x=(8n+1)\frac{\pi }{4},n\in Z$