1. Basic Reciprocal Identities
These identities directly relate each trigonometric ratio with the reciprocal of another. They come from the definitions in a right triangle.
\( \tan \theta = \dfrac{1}{\cot \theta} \)
\( \cot \theta = \dfrac{1}{\tan \theta} \)
\( \sec \theta = \dfrac{1}{\cos \theta} \)
\( \csc \theta = \dfrac{1}{\sin \theta} \)
2. Quotient Identities
tan and cot can be written using sin and cos, so these identities help rewrite expressions in simpler forms.
\( \tan \theta = \dfrac{\sin \theta}{\cos \theta} \)
\( \cot \theta = \dfrac{\cos \theta}{\sin \theta} \)
2.1. Example
If \( \sin \theta = \dfrac{3}{5} \) and \( \cos \theta = \dfrac{4}{5} \), then:
\( \tan \theta = \dfrac{3/5}{4/5} = \dfrac{3}{4} \)
\( \cot \theta = \dfrac{4/5}{3/5} = \dfrac{4}{3} \)
3. Pythagorean Identities Involving tan, sec, cot, cosec
These identities come from the fundamental Pythagorean identity by dividing with \(\cos^2 \theta\) or \(\sin^2 \theta\):
\( 1 + \tan^2 \theta = \sec^2 \theta \)
\( 1 + \cot^2 \theta = \csc^2 \theta \)
3.1. Example
If \( \tan \theta = 2 \), then:
\( \sec^2 \theta = 1 + \tan^2 \theta = 1 + 4 = 5 \)
\( \sec \theta = \sqrt{5} \)
4. Transforming tan and sec Identities
Using Pythagorean identities, we can rewrite expressions involving tan and sec:
- \( \sec^2 \theta - \tan^2 \theta = 1 \)
- \( \tan^2 \theta = \sec^2 \theta - 1 \)
These forms are helpful in simplification and equation solving.
4.1. Example
If \( \sec \theta = \dfrac{5}{4} \), then:
\( \sec^2 \theta = \dfrac{25}{16} \)
\( \tan^2 \theta = \sec^2 \theta - 1 = \dfrac{25}{16} - 1 = \dfrac{9}{16} \)
\( \tan \theta = \dfrac{3}{4} \)
5. Transforming cot and cosec Identities
Similarly, identities involving cot and cosec allow rearrangement:
- \( \csc^2 \theta - \cot^2 \theta = 1 \)
- \( \cot^2 \theta = \csc^2 \theta - 1 \)
5.1. Example
If \( \csc \theta = 5 \), then:
\( \csc^2 \theta = 25 \)
Using identity: \( \cot^2 \theta = \csc^2 \theta - 1 = 24 \)
\( \cot \theta = 2\sqrt{6} \)
6. Converting Between tan–sec and cot–cosec
Sometimes we rewrite tan in terms of sec, or cot in terms of cosec, to simplify an expression.
\( \tan \theta = \sqrt{\sec^2 \theta - 1} \)
\( \cot \theta = \sqrt{\csc^2 \theta - 1} \)