Trigonometry / Unit 6: Trigonometric Identities
6.3 Verifying Identities
Notes
To verify an identity, show that one side can be transformed into the other using known identities.
- Work on one side at a time (usually the more complicated one). Do not treat it like an equation to solve.
- Convert everything to sine and cosine, then combine fractions or multiply by a conjugate.
- Plugging in a number can disprove an identity, but it never proves one.
- Example: $\displaystyle \frac{\cos x}{1 + \sin x} \cdot \frac{1 - \sin x}{1 - \sin x} = \frac{\cos x (1 - \sin x)}{\cos^2 x} = \frac{1 - \sin x}{\cos x}$.
Practice
To verify $\tan x + \cot x = \sec x \csc x$, what is a good first step?
Converting to sine and cosine and combining fractions is a standard method. A single test value does not prove an identity.
To show $\displaystyle \frac{\cos x}{1 + \sin x} = \frac{1 - \sin x}{\cos x}$, multiply the numerator and denominator of the left side by:
The conjugate $1 - \sin x$ makes the denominator $1 - \sin^2 x = \cos^2 x$.
Which equation is NOT an identity?
Try $x = y = 90^\circ$: the left side is $\sin 180^\circ = 0$ but the right side is $2$.
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