Solve the following equations: + =2 - Vidyadip

Question

Solve the following equations: $\sin\text{x}+\cos\text{x}=\sqrt{2}$

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Answer

We have, $\sin\text{x}+\cos\text{x}=\sqrt{2}$ $\Rightarrow\frac{1}{\sqrt{2}}\sin\text{x}+\frac{1}{\sqrt{2}}\cos\text{x}=1$ $\Rightarrow\sin\frac{\pi}{4}\sin\text{x}+\cos\frac{\pi}{4}\cos\text{x}=1$$\Big[\because\cos\frac{\pi}{4}=\sin\frac{\pi}{4}=\frac{1}{\sqrt{3}}\Big]$ $\Rightarrow\cos\Big(\text{x}-\frac{\pi}{4}\Big)=\cos0^\circ$ $\Rightarrow\text{x}-\frac{\pi}{4}=2\text{n}\pi,\text{n}\in\text{z}$ $\Rightarrow\text{x}=2\text{n}\pi+\frac{\pi}{4},\text{n}\in\text{z}$ $\therefore\text{x}=(8\text{n}+1)\frac{\pi}{4},\text{n}\in\text{z}$

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