An identity holds for every angle on its common domain. A proof rewrites one side into the other using established identities; a few numerical checks do not prove it. From sin2x+cos2x=1, division by nonzero cos2x yields 1+tan2x=sec2x. This explains why reciprocal and quotient expressions can be reduced to sine and cosine. 1. State all original denominator restrictions. 2. Start with the more complicated side. 3. Rewrite quotients/reciprocals or factor. 4. Apply a Pythagorean identity and arrive at the other side. Example: sin2xsec2x=sin2x/cos2x=tan2x, where cos x 0. At x=pi/3, both sides equal 3; this checks the example, but the algebra proves it. Exam check: do not assume the target equality while proving it. A simplification never restores an angle excluded by the original denominator.

An identity holds for every angle on its common domain. A proof rewrites one side into the other using established identities; a few numerical checks do not prove it.

From \(\sin^2x+\cos^2x=1\), division by nonzero \(\cos^2x\) yields \(1+\tan^2x=\sec^2x\). This explains why reciprocal and quotient expressions can be reduced to sine and cosine.

1. State all original denominator restrictions. 2. Start with the more complicated side. 3. Rewrite quotients/reciprocals or factor. 4. Apply a Pythagorean identity and arrive at the other side.

Example: \(\sin^2x\sec^2x=\frac{\sin^2x}{\cos^2x}=\tan^2x\), where \(\cos x\ne0\). At \(x=\frac{\pi}{3}\), both sides equal \(3\); this checks the example, but the algebra proves it.

Exam check: do not assume the target equality while proving it. A simplification never restores an angle excluded by the original denominator.

One-sided identity proofs

One-sided identity proofs