Why does cos(90 - x) = sin(x) and sin(90 - x) = cos(x)?

The identities \(\cos(90^\circ - x) = \sin(x)\) and \(\sin(90^\circ - x) = \cos(x)\) are based on the co-function identities in trigonometry. These identities hold true because of the relationship between the angles in a right-angled triangle and the unit circle.

Here's a detailed explanation:

1. **Consider a right-angled triangle:**
- In a right-angled triangle, the angles add up to 90° and 180°. Let one of the angles be \(x\), so the other non-right angle will be \(90^\circ - x\).
- The trigonometric functions for an angle in a right-angled triangle are defined as:
- \(\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

2. **In terms of complementary angles:**
- When you consider the angle \(90^\circ - x\), the opposite and adjacent sides switch roles. For example:
- The side opposite to angle \(90^\circ - x\) is the same as the side adjacent to angle \(x\).
- The side adjacent to angle \(90^\circ - x\) is the same as the side opposite to angle \(x\).

This means:
- \(\cos(90^\circ - x)\) becomes the side opposite \(x\) (which is \(\sin(x)\)).
- \(\sin(90^\circ - x)\) becomes the side adjacent to \(x\) (which is \(\cos(x)\)).

3. **Unit Circle Explanation:**
- The unit circle provides a geometric interpretation of the trigonometric functions. For any angle \(\theta\), the coordinates of a point on the unit circle are \((\cos(\theta), \sin(\theta))\).
- For the angle \(90^\circ - x\), the coordinates of the point on the unit circle are \((\cos(90^\circ - x), \sin(90^\circ - x))\).
- Using the co-function identities:
- \(\cos(90^\circ - x) = \sin(x)\)
- \(\sin(90^\circ - x) = \cos(x)\)

In conclusion, these identities come from the complementary relationship between the angles and the switching of the opposite and adjacent sides in a right triangle, or from the geometric interpretation on the unit circle.