What is the co-function identity relating \( \tan\) and \( \cot\)?

The co-function identity relating the tangent (\(\tan\)) and cotangent (\(\cot\)) functions involves complementary angles. Complementary angles are two angles whose measures add up to \(90^\circ\) (or \(\frac{\pi}{2}\) radians). **Identity:** \[ \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \] **Explanation:** - **\(\theta\):** This represents an angle in a right triangle. - **\(\frac{\pi}{2} - \theta\):** This is the complementary angle to \(\theta\). - **\(\tan\left(\frac{\pi}{2} - \theta\right)\):** The tangent of the complementary angle. - **\(\cot(\theta)\):** The cotangent of the original angle \(\theta\). **Example:** If \(\theta = 30^\circ\), then: \[ \tan\left(90^\circ - 30^\circ\right) = \tan(60^\circ) = \sqrt{3} \] \[ \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \sqrt{3} \] Thus, \(\tan(60^\circ) = \cot(30^\circ)\), verifying the identity. **Graphical Interpretation:** On the unit circle, the tangent of an angle and the cotangent of its complementary angle are equal because they represent the slopes of lines that are perpendicular to each other. **Summary:** The co-function identity demonstrates a fundamental relationship between \(\tan\) and \(\cot\) based on complementary angles, highlighting the interconnected nature of trigonometric functions. **Final Identity:** \[ \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \] Answer: For all angles θ, the tangent of (π⁄2 – θ) equals cot θ. In symbols, tan(π/2 − θ) = cotθ