Key takeaway: Trigonometric identities are equations between trig expressions that hold for every angle where both sides are defined, used to simplify expressions and prove new relationships.
Grade 12 – PrecalculusF-TF.C.9Trigonometry
Definition of Trigonometric Identities
Trigonometric identities are statements that stay true no matter which angle you plug in, as long as both sides make sense. They are different from conditional equations such as \(\sin x=\frac{1}{2}\), which hold only for certain angles.
The basic ones fall into four families: reciprocal, quotient, Pythagorean and even-odd. With these as raw material, you can rewrite a messy expression in a simpler form, switch everything to sines and cosines, or prove that two different-looking expressions are equal. Proofs work on one side at a time and never move terms across the equals sign.
Formula and notation
The core list of trigonometric identities, for angles where each side is defined:
\[\csc x=\frac{1}{\sin x},\quad \tan x=\frac{\sin x}{\cos x},\quad \sin^2x+\cos^2x=1,\quad \cos(-x)=\cos x,\quad \sin(-x)=-\sin x.\]
Dividing the Pythagorean one by \(\cos^2x\) gives \(1+\tan^2x=\sec^2x\).
Key facts
- To prove one of the trigonometric identities, transform one side until it matches the other.
- Rewriting everything in sine and cosine is the most reliable first move.
- A single numerical test cannot prove an identity, but can disprove one.
- Cosine is an even function; sine and tangent are odd.
Trigonometric Identities example
Simplify \((1+\tan^2\theta)\cos^2\theta\) to a single number.
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Replace \(1+\tan^2\theta\) with \(\sec^2\theta\).
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Rewrite the secant: \(\sec^2\theta=\frac{1}{\cos^2\theta}\).
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Multiply: \(\frac{1}{\cos^2\theta}\cdot\cos^2\theta=1\).
The expression always equals 1, wherever \(\cos\theta\neq 0\).
Common mistake
Watch out
Alina cancels \(\sin x\) in \(\frac{\sin x}{\sin x+\cos x}\) and writes \(\frac{1}{1+\cos x}\).
Why it is wrong
Only a factor of the whole top and bottom may cancel, and \(\sin x\) in the bottom is one term of a sum. Test \(x=30^{\circ}\): the original is about 0.366, while Alina’s result is about 0.536.
Related terms and lessons
- Pythagorean Identity
- Double-Angle Formulas
- Sum and Difference Formulas
- Unit Circle
- Inverse Trigonometric Functions
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