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Question Number 180407 by CrispyXYZ last updated on 12/Nov/22

$$f(x-y)+f(x+y)=f\left(x\right)f\left(y\right)$$$$\mathrm{find}f\left(x\right)$$

Answered by mahdipoor last updated on 12/Nov/22

$$x=0y=0\Rightarrow$$$$f\left(0\right)+f\left(0\right)=f\left(0\right)f\left(0\right)\Rightarrow f\left(0\right)=0or2$$$$x=xy=0\Rightarrow$$$$f\left(x\right)+f\left(x\right)=f\left(0\right)f\left(x\right)\Rightarrow$$$$\{\begin{array}{l}iff\left(0\right)=0\Rightarrow f\left(x\right)=0\mathrm{\forall }x\\ iff\left(0\right)=2\Rightarrow f\left(x\right)=0\mathrm{\forall }x\stackrel{but}{\Rightarrow }f\left(0\right)=2and0\Rightarrow rejected\end{array}$$

Answered by Rasheed.Sindhi last updated on 12/Nov/22

$$f(x-y)+f(x+y)=f\left(x\right)f\left(y\right)...\left(i\right)$$$$f\left(x\right)=?$$$$\bullet y=0in\left(i\right):$$$$2f\left(x\right)=f\left(x\right)f\left(0\right)\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(ii\right)$$$$2f\left(x\right)-f\left(x\right)f\left(0\right)=0$$$$f\left(x\right)(2-f\left(0\right))=0$$$$f\left(x\right)=0\vee f\left(0\right)=2$$$$\bullet f\left(0\right)=2in\left(ii\right):$$$$2f\left(x\right)=2f\left(x\right)\to nousefulresult$$$$\mathcal{H}ence$$$$\overline{)\begin{array}{c}f\left(x\right)=0\end{array}}$$

Commented by CrispyXYZ last updated on 13/Nov/22

$$\mathrm{but}\mathrm{consider}f\left(x\right)=2\cos x$$$$left=2\cos (x-y)+2\cos (x+y)$$$$=2(\cos (x-y)+\cos (x+y))$$$$=4\cos x\cos y$$$$=right$$

Commented by CrispyXYZ last updated on 13/Nov/22

$$\mathrm{Actually},f\left(x\right)=0\vee f\left(x\right)=2\vee f\left(x\right)=2\cos x\mathrm{satisfy}\mathrm{the}\mathrm{condition}$$

Commented by Rasheed.Sindhi last updated on 13/Nov/22

$$\mathcal{T}han\mathcal{X}!\mathcal{B}utnowthequestionis$$$${what}^{\prime }stheprocesstoreachf\left(x\right)=2\cos \left(x\right)$$$$thatiswhyI{}^{\prime }vepostedamoredirect$$$$question:$$$$f(x-y)+f(x+y)=2f\left(x\right)f\left(y\right);f\left(x\right)=?$$$$theanswerisf\left(x\right)=\cos \left(x\right)$$

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