The Value of \(2\sin^2B + 4\cos(A+B)\sin A\sin B + \cos2(A+B)\)
Question
Find the value of
\[ 2\sin^2B + 4\cos(A+B)\sin A\sin B + \cos2(A+B) \]
(a) \(0\)
(b) \(\cos3A\)
(c) \(\cos2A\)
(d) none of these
Solution
Use the identity
\[ 2\sin^2B = 1-\cos2B \]
and
\[ \cos2(A+B)=2\cos^2(A+B)-1 \]
Substituting,
\[ =1-\cos2B +4\cos(A+B)\sin A\sin B +2\cos^2(A+B)-1 \]
\[ =2\cos^2(A+B)-\cos2B +4\cos(A+B)\sin A\sin B \]
Using
\[ 2\cos^2(A+B)=1+\cos2(A+B) \]
and simplifying through product-to-sum identities,
\[ = \cos2A \]
A shorter verification can be obtained by expanding \(\cos2(A+B)\) and combining terms using \(\cos(A+B)=\cos A\cos B-\sin A\sin B\). All intermediate terms cancel, leaving
\[ \cos2A. \]
Final Answer
\[ \boxed{\cos2A} \]
Hence, the correct option is (c) \(\cos2A\).