Write the value of \( \dfrac{\sin A+\sin3A}{\cos A+\cos3A} \)
Solution:
Using identities,
\[
\sin C+\sin D
=
2\sin\frac{C+D}{2}\cos\frac{C-D}{2}
\]
\[
\sin A+\sin3A
=
2\sin2A\cos A
\]
Also,
\[
\cos C+\cos D
=
2\cos\frac{C+D}{2}\cos\frac{C-D}{2}
\]
\[
\cos A+\cos3A
=
2\cos2A\cos A
\]
Therefore,
\[
\frac{\sin A+\sin3A}{\cos A+\cos3A}
=
\frac{2\sin2A\cos A}{2\cos2A\cos A}
\]
\[
=
\tan2A
\]
\[
\boxed{\tan2A}
\]