Find the Value of cos(36° − A) cos(36° + A) + cos(54° + A) cos(54° − A)

Solution

Using the identity:

\[ \cos(C-D)\cos(C+D) = \cos^2C-\sin^2D \]

Therefore,

\[ \cos(36^\circ-A)\cos(36^\circ+A) = \cos^236^\circ-\sin^2A \]

Also,

\[ \cos(54^\circ+A)\cos(54^\circ-A) = \cos^254^\circ-\sin^2A \]

Adding,

\[ = \cos^236^\circ+\cos^254^\circ-2\sin^2A \]

Since

\[ 54^\circ=90^\circ-36^\circ \]

we get

\[ \cos54^\circ=\sin36^\circ \]

Therefore,

\[ \cos^236^\circ+\cos^254^\circ = \cos^236^\circ+\sin^236^\circ = 1 \]

Hence,

\[ 1-2\sin^2A \]

Using the identity:

\[ \cos2A=1-2\sin^2A \]

Therefore,

\[ \boxed{ \cos(36^\circ-A)\cos(36^\circ+A) + \cos(54^\circ+A)\cos(54^\circ-A) = \cos2A } \]

Final Answer

\[ \boxed{ \cos(36^\circ-A)\cos(36^\circ+A) + \cos(54^\circ+A)\cos(54^\circ-A) = \cos2A } \]

Correct Option: (b)