Question

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

• A. Dependent on B
• B. Dependent on A and B
• C. Dependent on A
• D. Independent of A and B

Answer 1

Answer: (C)

Dependent on A

Answer 2

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

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

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

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

Hence it depends on A.

Trick : Put two different values of A.

Let $A = 90^{o},$ then the value of expression will be

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

Now put $A = 0^{o}$, then the value of expression will be

$\cos^{2}B + \cos^{2}B - 2\cos^{2}B = 0$

It means that the expression has different values for different

A i.e. it depends on A.

Now similarly for $B = 90^{o},$ the value of expression will be

$\sin^{2}A + 0 - 0 = \sin^{2}A$ and at $B = 0^{o}$ the value of expression

will be $\cos^{2}A + 1 - 2\cos^{2}A = \sin^{2}A$.

Hence, the expression has the same value for different values

of B, so it does not depend on B.