Question

Solve the equation
$\left(iii\right)\sin x+\cos x=1$

Answer 1

$\sin x+\cos x=1$

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

$\Rightarrow \sin \frac{\pi }{4}\sin +\cos \frac{\pi }{4}\cos x=\frac{1}{\sqrt{2}}$

$\Rightarrow \cos (x-\frac{\pi }{4})=\frac{1}{\sqrt{2}}$

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

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

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

So,

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

On taking positive sign, we get

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

$\Rightarrow x=2n\pi +\frac{\pi }{2}=(4n+1)\frac{\pi }{2}$

On taking negative sign, we get

$\Rightarrow (x-\frac{\pi }{4})=2n\pi -\frac{\pi }{4}$

$\Rightarrow x=2n\pi$

Hence, the general is
$x=(4n+1)\frac{\pi }{2}\text{and}x=2n\pi ,n\in Z$