Prove that: \[ \sin 65^\circ + \cos 65^\circ = \sqrt{2}\cos 20^\circ \]
Solution
Using the identity:
\[
\sin \theta + \cos \theta
=
\sqrt{2}\cos(45^\circ-\theta)
\]
Taking
\[
\theta = 65^\circ
\]
Then,
\[
\sin 65^\circ + \cos 65^\circ
=
\sqrt{2}\cos(45^\circ-65^\circ)
\]
\[
=
\sqrt{2}\cos(-20^\circ)
\]
Since,
\[
\cos(-\theta)=\cos\theta
\]
\[
=
\sqrt{2}\cos20^\circ
\]
Hence,
\[
\boxed{
\sin 65^\circ + \cos 65^\circ
=
\sqrt{2}\cos20^\circ
}
\]