Question

The expression ${\cos }^2\varphi +{\cos }^2(a+\varphi )-2\cos a\cos \varphi \cos (a+\varphi )$ is independent is

• A. $\varphi$
• B. $a$
• C. both $a$ and $\varphi$
• D. none of $a$ and $\varphi$

Answer 1

The correct option is A $\varphi$

The given expression is equal to
${\cos }^2\varphi +{\cos }^2(a+\varphi )-\left[\cos \right(a+\varphi )+\cos (a-\varphi \left)\right]\cos (a+\varphi )$
$={\cos }^2\varphi -\cos (a+\varphi )\cos (a-\varphi )$
$={\cos }^2\varphi -({\cos }^2\varphi -{\sin }^{2}a)={\sin }^{2}a$
which is independent of $\varphi$.