Final answer:
The complicated math equation that equals 1 is option C, sin²θ + cos²θ = 1, thus the correct option is C.
Explanation:
In this equation, we are dealing with trigonometric functions, specifically sine and cosine. Before we dive into the explanation, let's define what sine and cosine are. Sine (sin) is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. Cosine (cos) is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Now, back to the equation. We can see that it is in the form of sin²θ + cos²θ = 1. This is known as the Pythagorean identity, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
To further understand this, let's take a look at an example. Consider a right-angled triangle with an angle θ. The opposite side is represented by a, the adjacent side is represented by b, and the hypotenuse is represented by c. According to the Pythagorean theorem, we have a² + b² = c². Now, if we divide both sides of the equation by c², we get (a²/c²) + (b²/c²) = 1. Remember that sin is equal to a/c and cos is equal to b/c. So, we can rewrite the equation as sin²θ + cos²θ = 1.
This means that for any given angle θ, the square of the sine of that angle plus the square of the cosine of that angle is always equal to 1. This holds true for all angles, whether acute, obtuse, or right angles.
In conclusion, the complicated math equation that equals 1, sin²θ + cos²θ = 1, is a fundamental identity in trigonometry known as the Pythagorean identity. It states that the square of the sine of an angle plus the square of the cosine of that angle is always equal to 1. This can be further explained by the Pythagorean theorem, which relates the sides of a right-angled triangle, thus the correct option is C.