Sine and cosine lie between $-1$ and $1$

How to use it

1. 1If a sum of several such terms reaches its largest possible total, every term must sit at its own maximum - there is no slack to trade.
2. 2Use it to reject an equation outright: anything forcing $\sin \theta$ or $\cos \theta$ beyond $1$ has no solution.
3. 3Pair it with ${\sin }^2\theta +{\cos }^2\theta =1$ to turn a cosine pinned at $\pm 1$ straight into a sine of $0$.

Step-by-step

1. 1
   $\cos \theta \le 1$ and $\cos 2\theta \le 1$, so the sum is at most $2$

   Why: Each term is separately bounded above by $1$.

2. 2
   $\text{sum}=2\Rightarrow \cos \theta =1\text{ and }\cos 2\theta =1\Rightarrow \theta =0^{\circ }$

   Why: Equality forces every term to its maximum at once.