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Question Number 105892 by mohammad17 last updated on 01/Aug/20

$$provethatsin\left(x\right)+cos\left(x\right)=\sqrt{2}cos(x-\frac{\pi }{4})?$$

Answered by john santu last updated on 01/Aug/20

$$\sin x+\cos x=p$$$$\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\frac{p}{\sqrt{2}}$$$$\Rightarrow \sin \left(\frac{\pi }{4}\right)\sin x+\cos \left(\frac{\pi }{4}\right)\cos x$$$$=\frac{p}{\sqrt{2}}$$$$(recall:\cos t\cos h+\sin t\sin h$$$$=\cos (t-h))$$$$\iff \cos (x-\frac{\pi }{4})=\frac{p}{\sqrt{2}}$$$$p=\sqrt{2}\cos (x-\frac{\pi }{4}).$$$$proved.$$

Commented by mohammad17 last updated on 01/Aug/20

$$thankyousir$$

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