The Value of \( \cos(36^\circ-A)\cos(36^\circ+A)+\cos(54^\circ-A)\cos(54^\circ+A) \)
Question
Find the value of
\[ \cos(36^\circ-A)\cos(36^\circ+A) + \cos(54^\circ-A)\cos(54^\circ+A) \]
(a) \(\cos2A\)
(b) \(\sin2A\)
(c) \(\cos A\)
(d) \(0\)
Solution
Use the identity
\[ \cos(x-y)\cos(x+y) = \cos^2x-\sin^2y \]
Therefore,
\[ \cos(36^\circ-A)\cos(36^\circ+A) = \cos^236^\circ-\sin^2A \]
and
\[ \cos(54^\circ-A)\cos(54^\circ+A) = \cos^254^\circ-\sin^2A \]
Adding,
\[ = \cos^236^\circ+\cos^254^\circ-2\sin^2A \]
Since
\[ 54^\circ=90^\circ-36^\circ \]
\[ \cos^254^\circ = \sin^236^\circ \]
Hence,
\[ \cos^236^\circ+\sin^236^\circ = 1 \]
Therefore,
\[ 1-2\sin^2A \]
Using the identity
\[ \cos2A=1-2\sin^2A \]
we get
\[ \cos(36^\circ-A)\cos(36^\circ+A) + \cos(54^\circ-A)\cos(54^\circ+A) = \cos2A \]
Final Answer
\[ \boxed{\cos2A} \]
Hence, the correct option is (a) \(\cos2A\).