Question

The expression ${\cos }^2(A-B)+{\cos }^{2}B-2\cos (A-B)\cos A\cos B$ is

• A. dependent on $B$ only.
• B. dependent on $A$ and $B$.
• C. dependent on $A$ only.
• D. independent of $A$ and $B$.

Answer 1

The correct option is C dependent on $A$ only.

${\cos }^2(A-B)+{\cos }^{2}B-2\cos (A-B)\cos A\cos B$

$\cos (A-B)\left[\cos \right(A-B)-2\cos A\cos B]+{\cos }^{2}B$

$=\cos (A-B)[\sin A\sin B-\cos A\cos B]+{\cos }^{2}B$

$=-\cos (A-B)\cos (A+B)+{\cos }^{2}B$

$=-\left[\frac{\cos 2A+\cos 2B}{2}\right]+{\cos }^{2}B$
$=-\left[\frac{2({\cos }^{2}A+{\cos }^{2}B)-2}{2}\right]+{\cos }^{2}B$

$=-{\cos }^{2}A-{\cos }^{2}B+1+{\cos }^{2}B$
$=1-{\cos }^{2}A$
$={\sin }^{2}A$

Expression is only dependent of A.