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Math101learn.math101.caTrigonometric Identities
Trigonometric identities are equations true for every input where both sides are defined and can simplify, verify, or transform expressions.
An identity does not find one angle; it proves two expressions describe the same quantity throughout their shared domain.
Identity versus conditional equation
An identity such as
is true for every real $\theta$. A conditional equation such as $\sin\theta=1/2$ is true only for selected angles.
When verifying an identity, the goal is to transform one or both expressions into the same form without assuming what must be proved.
Reciprocal identities
The reciprocal functions satisfy
where denominators are nonzero. Reciprocal identities are useful for replacing less familiar functions with sine and cosine.
Quotient identities
Tangent and cotangent can be written as
Converting everything to sine and cosine often exposes common factors or a Pythagorean pattern.
Pythagorean identities
The central identity is
Dividing it by $\cos^2\theta$ gives
and dividing by $\sin^2\theta$ gives
Rearranged forms such as $1-\cos^2\theta=\sin^2\theta$ are common in verification.
Worked example: verify an identity
A structured verification strategy
- Work on the more complicated side first.
- Factor before expanding when possible.
- Replace reciprocal and quotient functions.
- Use a common denominator for sums of fractions.
- Look for Pythagorean expressions.
- Stop once the other side appears exactly.
Write one justified equality per line so the logic can be checked.
Common-denominator example
To simplify
use the product denominator:
The conjugate factors create a Pythagorean difference.
Factoring trigonometric expressions
Treat $\sin\theta$ or $\cos\theta$ as algebraic quantities when appropriate:
This supports both simplification and solving. Normal factoring rules still apply.
Domain awareness
An identity is true wherever both original sides are defined. Cancelling a factor can hide exclusions. For example, $\sin^2\theta/\sin\theta=\sin\theta$ requires $\sin\theta\ne0$ in the original quotient.
Do not claim a larger domain merely because the simplified expression is defined there.
Verification is not circular proof
Avoid starting by writing the two sides equal and performing operations across the equality, because that assumes the conclusion. Label one side and transform it independently, or transform both sides separately into a common expression.
Testing a few angles can detect a false claim but cannot prove an identity for infinitely many inputs.
Common mistakes
Cancelling across addition. Only common factors can cancel.
Replacing $\sin^2+\cos^2$ with the wrong value. It equals $1$.
Applying an identity to the wrong structure. For example, $1-\cos\theta$ is not $\sin\theta$.
Manipulating both sides as if solving. Verify through valid equivalent rewrites.
Ignoring original restrictions. Cancelled denominators still affect the shared domain.
Quick self-check
- Is the statement intended as an identity or an equation to solve?
- Which side is structurally more complicated?
- Can reciprocal or quotient functions become sine and cosine?
- Is there a Pythagorean pattern, factor, or common denominator?
- Have I preserved original domain restrictions?
- Does every line follow from a named or visible algebraic step?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Simplify (1 − cos² θ)/sin θ where the original expression is defined.
- Replace the numerator with sin² θ.
- sin² θ/sin θ = sin θ when sin θ ≠ 0.
End of lesson
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