How To Solve Equations Using Trigonometric Identities: A Step-By-Step Guide

sin^2x + cos^2x =

1

The equation sin^2x + cos^2x = 1 is known as the Pythagorean identity. This identity holds true for all values of x.

To understand why this identity is true, we can use the unit circle. If we draw a unit circle (a circle with a radius of 1) and place it on a coordinate plane, the coordinates of any point on the circle can be written as (cosθ, sinθ), where θ is the angle formed by the terminal side of x and the positive x-axis.

Now, let’s consider the coordinates of a point on the unit circle. We know that the distance from the origin to the point is 1, so we can use the Pythagorean theorem to write:

(cosθ)^2 + (sinθ)^2 = 1

In other words, the sum of the squares of the cosine and sine of an angle is always equal to 1. This relationship is true for all angles, so we can say that:

sin^2x + cos^2x = 1

Therefore, whenever you see sin^2x + cos^2x in an equation, you can simplify it to 1.

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