Final answer:
The identity Cos(t)² + Sin(t)² = 1 is known as the Pythagorean trigonometric identity and can be derived using the Pythagorean Theorem in the context of a unit circle.
Step-by-step explanation:
The question is about proving the trigonometric identity Cos(t)² + Sin(t)² = 1. This is a fundamental identity in trigonometry, commonly referred to as the Pythagorean trigonometric identity. It can be derived from the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
If we consider a unit circle (a circle with a radius of 1) where a point on the circumference has coordinates (Cos(t), Sin(t)), by the definition of sine and cosine in a circle of radius 1, these coordinates are the lengths of the adjacent and opposite sides of a right triangle inscribed in the circle. The hypotenuse in this case is the radius of the circle, which is 1. According to the Pythagorean Theorem: Adjacent side² + Opposite side² = Hypotenuse², which translates to Cos(t)² + Sin(t)² = 1.