How do I verify a trigonometric identity without assuming the two sides are equal?

Work on just one side at a time. Choose the side that looks more complicated, then rewrite it using established identities and algebra until it becomes the other side. Each line should follow from the line before it by a valid identity or algebraic step—not by assuming the proposed equality is already true. Factoring, expansion, and finding common denominators are possible algebraic tools [S1].

Keep the other side visible as a target, but don’t use it to justify a step. If your transformations reach that target, you have a proof wherever the original expressions are defined. The domain check matters: a transformation involving division can introduce a restriction that must not be overlooked.

Take the identity \(1+\cot^2\theta=\csc^2\theta\). Start with its left side and use \(\cot\theta=\frac{\cos\theta}{\sin\theta}\) and \(\sin^2\theta+\cos^2\theta=1\):

\[ \begin{aligned} 1+\cot^2\theta &=1+\frac{\cos^2\theta}{\sin^2\theta}\\ &=\frac{\sin^2\theta+\cos^2\theta}{\sin^2\theta}\\ &=\frac{1}{\sin^2\theta}\\ &=\csc^2\theta. \end{aligned} \]

The first and last expressions already require \(\sin\theta\ne0\), because cotangent and cosecant have sine in the denominator. The fraction in the derivation has that same restriction. Thus the identity holds on the domain where the original expressions exist; it is not a claim about angles with \(\sin\theta=0\) [S1].

For every step, ask what made it legal. Replacing \(1-\cos^2\theta\) with \(\sin^2\theta\) uses the Pythagorean identity. Combining fractions needs a common denominator. Canceling a factor is valid only when that factor is nonzero on the domain you are considering.

Also distinguish the original domain from restrictions introduced by your working. Write down the denominators in the original expressions first. Then note any extra nonzero assumptions made during transformations. If an extra restriction was not part of the original problem, you may have proved the result only on a smaller set; check whether the original expressions are defined at the omitted angles and, if so, handle those angles separately.

Try this routine:

1. Record where each original side is defined by checking its denominators.
2. Choose one side to transform, usually the more complicated one.
3. Make one justified change per line, naming the identity or algebra used if helpful.
4. Check every division or cancellation for a nonzero condition.
5. Stop when the transformed expression matches the untouched other side, and state the domain of the result.

For your written solution, a useful final sentence is: “These steps establish the identity for all \(\theta\) in the domain of the original expressions.” Add the actual domain restriction for the problem. If the expressions do not simplify to the target, that does not by itself prove the statement false; it means this attempt has not verified it.