how to proof that the following one is true
$\sin2 \theta + \cos2 \theta = \sin \theta + \cos \theta$
I tried to do like this
L.H.S.
$= 2\sin\theta\cos\theta + \cos^2 \theta – \sin^2 \theta $
$= \sin\theta\cos\theta + \cos^2 \theta + \sin\theta\cos\theta – \sin^2 \theta$
$= \cos \theta(\sin\theta + \cos \theta) + \sin \theta(\cos\theta – \sin \theta)$
Then what should I do ?
Am I on the right way ?
Best Answer
Let $(*)$ be the equation to solve.
$$ (*) \Leftrightarrow \sqrt2 \cos(2\theta-\frac{\pi}{4}) = \sqrt2 \cos(\theta-\frac{\pi}{4})$$
$$\Leftrightarrow 2\theta-\frac{\pi}{4} \equiv \theta-\frac{\pi}{4} \pmod{2\pi} \text{ or } 2\theta-\frac{\pi}{4} \equiv -\theta+\frac{\pi}{4} \pmod{2\pi}$$
$$ \Leftrightarrow \theta \equiv 0 \pmod{2\pi} \text{ or } \theta \equiv \frac{\pi}{6} \pmod{\frac{2\pi}{3}}$$