Find all (a,b), such that $y={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}(x+a)+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$ is constant for all $x\in \mathbb{R}$

Gretchen Schwartz
2022-07-06
Answered

Find all (a,b), such that $y={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}(x+a)+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$ is constant for all $x\in \mathbb{R}$

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Asdrubali2r

Answered 2022-07-07
Author has **14** answers

HINT:

$S={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}(x+a)+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$

$=1+{\mathrm{cos}}^{2}(x+a)-{\mathrm{sin}}^{2}x+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$

Using Prove that $\mathrm{cos}(A+B)\mathrm{cos}(A-B)={\mathrm{cos}}^{2}A-{\mathrm{sin}}^{2}B$

$S=1+\mathrm{cos}a\mathrm{cos}(2x+a)+\{\mathrm{cos}(2x+a)+\mathrm{cos}a\}\mathrm{cos}b$

$=1+\mathrm{cos}a\mathrm{cos}b+(\mathrm{cos}b+\mathrm{cos}a)\mathrm{cos}(2x+a)$

$S={\mathrm{cos}}^{2}x+{\mathrm{cos}}^{2}(x+a)+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$

$=1+{\mathrm{cos}}^{2}(x+a)-{\mathrm{sin}}^{2}x+2\mathrm{cos}x\mathrm{cos}(x+a)\mathrm{cos}b$

Using Prove that $\mathrm{cos}(A+B)\mathrm{cos}(A-B)={\mathrm{cos}}^{2}A-{\mathrm{sin}}^{2}B$

$S=1+\mathrm{cos}a\mathrm{cos}(2x+a)+\{\mathrm{cos}(2x+a)+\mathrm{cos}a\}\mathrm{cos}b$

$=1+\mathrm{cos}a\mathrm{cos}b+(\mathrm{cos}b+\mathrm{cos}a)\mathrm{cos}(2x+a)$

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