Identities

The key idea is to rewrite every trigonometric function in terms of $\sin \theta$ and $\cos \theta$ using $\tan \theta =\frac{\sin \theta }{\cos \theta }$, $\sec \theta =\frac{1}{\cos \theta }$, $\cot \theta =\frac{\cos \theta }{\sin \theta }$, and $\csc \theta =\frac{1}{\sin \theta }$. Then clear denominators and apply ${\sin }^2\theta +{\cos }^2\theta =1$ to simplify. This single move proves most identities and turns products into manageable expressions. For products like $\sin \theta \sin (60^{\circ }-\theta )\sin (60^{\circ }+\theta )$, expand the shifted sines with $\sin (a\pm b)=\sin a\cos b\pm \cos a\sin b$ before simplifying.