What is the co-function identity relating $$ an$$ and $$ \cot$$?

Question

What is the co-function identity relating $$ 	an$$ and $$ \cot$$?

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The **co-function identities** in trigonometry relate the trigonometric functions of complementary angles (angles that add up to $$90^\circ$$ or $$\frac{\pi}{2}$$ radians). Specifically, for the tangent and cotangent functions, the co-function identity is:

$$
\tan\left(90^\circ - \theta\right) = \cot \theta
$$
$$
\tan\left(\frac{\pi}{2} - x\right) = \cot x
$$

### Explanation:

- **Degrees Version:**
  $$
  \tan(90^\circ - \theta) = \cot \theta
  $$
  This means that the tangent of the complement of an angle $$\theta$$ is equal to the cotangent of $$\theta$$.

- **Radians Version:**
  $$
  \tan\left(\frac{\pi}{2} - x\right) = \cot x
  $$
  Similarly, in radians, the tangent of the complement of angle $$x$$ (where $$\frac{\pi}{2}$$ radians is $$90^\circ$$) is equal to the cotangent of $$x$$.

### Visual Representation:

Consider a right triangle where one of the acute angles is $$\theta$$. The other acute angle will then be $$90^\circ - \theta$$ (since the sum of angles in a triangle is $$180^\circ$$ and one angle is $$90^\circ$$).

- **Tangent of the complementary angle:**
  $$
  \tan(90^\circ - \theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \cot \theta
  $$
  
### Applications:

- **Simplifying Expressions:** Co-function identities are useful in simplifying trigonometric expressions and solving equations involving complementary angles.
- **Integration and Differentiation:** These identities can be helpful in calculus when integrating or differentiating trigonometric functions.
- **Solving Triangles:** In geometry and trigonometry problems, especially those involving right triangles, co-function identities help relate different trigonometric ratios.

### Summary:

The co-function identity connecting tangent and cotangent is a fundamental relationship in trigonometry that expresses how these two functions are complementary to each other. Remembering that $$\tan(90^\circ - \theta) = \cot \theta$$ (or in radians, $$\tan\left(\frac{\pi}{2} - x\right) = \cot x$$) can simplify many trigonometric problems.

**Formula:**
$$
\tan\left(90^\circ - \theta\right) = \cot \theta \quad \text{or} \quad \tan\left(\frac{\pi}{2} - x\right) = \cot x
$$
The co-function identity relating $$ \tan $$ and $$ \cot $$ is: $$ \tan\left(90^\circ - \theta\right) = \cot \theta $$ $$ \tan\left(\frac{\pi}{2} - x\right) = \cot x $$ This means the tangent of the complement of an angle is equal to the cotangent of that angle.

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

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