Question

The expression
$f(\alpha ,\beta )={\cos }^2\alpha +{\cos }^2(\alpha +\beta )-2\cos \alpha \cos \beta \cos (\alpha +\beta )$

• A. is independent of $\beta$
• B. is independent of $\alpha$
• C. is independent of both $\alpha$ and $\beta$
• D. is dependent of $\beta$

Answer 1

Answer: (B)

is independent of $\alpha$

$f(\alpha ,\beta )={\cos }^2\alpha +{\cos }^2(\alpha +\beta )-2\cos \alpha \cos \beta \cos (\alpha +\beta )$

$={\cos }^2\alpha +\cos (\alpha +\beta )\left[\cos \right(\alpha +\beta )-2\cos \alpha \cos \beta ]$

$={\cos }^2\alpha +\cos (\alpha +\beta )[\cos \alpha \cos \beta -\sin \alpha \sin \beta -2\cos \alpha \cos \beta ]$

$={\cos }^2\alpha -\cos (\alpha +\beta )[\sin \alpha \sin \beta +\cos \alpha \cos \beta ]$

$={\cos }^2\alpha -\cos (\alpha +\beta ).\cos (\alpha -\beta )$

$={\cos }^2\alpha -({\cos }^2\alpha -{\sin }^2\beta )$

$={\sin }^2\beta$

$\therefore$ it is independent of $\alpha$.