# Find all (a,b), such that y = cos 2 </msup> &#x2061;<!-- ⁡ --> x +

Find all (a,b), such that $y={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}\left(x+a\right)+2\mathrm{cos}x\mathrm{cos}\left(x+a\right)\mathrm{cos}b$ is constant for all $x\in \mathbb{R}$
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Asdrubali2r
HINT:
$S={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}\left(x+a\right)+2\mathrm{cos}x\mathrm{cos}\left(x+a\right)\mathrm{cos}b$
$=1+{\mathrm{cos}}^{2}\left(x+a\right)-{\mathrm{sin}}^{2}x+2\mathrm{cos}x\mathrm{cos}\left(x+a\right)\mathrm{cos}b$
Using Prove that $\mathrm{cos}\left(A+B\right)\mathrm{cos}\left(A-B\right)={\mathrm{cos}}^{2}A-{\mathrm{sin}}^{2}B$
$S=1+\mathrm{cos}a\mathrm{cos}\left(2x+a\right)+\left\{\mathrm{cos}\left(2x+a\right)+\mathrm{cos}a\right\}\mathrm{cos}b$
$=1+\mathrm{cos}a\mathrm{cos}b+\left(\mathrm{cos}b+\mathrm{cos}a\right)\mathrm{cos}\left(2x+a\right)$