sin^2x + cos^2x =
1
The equation sin^2x + cos^2x = 1 is a fundamental trigonometric identity. It states that for any angle x, the sum of the squares of the sine and cosine of that angle is always equal to 1. This identity can be derived in various ways, but one of the most common methods is using the Pythagorean identity.
The Pythagorean identity states that sin^2x + cos^2x = 1 – sin^2x. To prove this identity, we start with the unit circle, which is a circle with a radius of 1 centered at the origin of the xy-plane. Any point on the unit circle can be represented by the coordinates (cos x, sin x), where x is the angle between the positive x-axis and the line connecting the origin to the point.
Now, draw a line connecting the point (cos x, sin x) to the x-axis, as shown in the figure below.
This line is perpendicular to the x-axis, so it forms a right triangle with the x-axis and the radius of the unit circle, as shown below.
By the Pythagorean theorem, we have:
hypotenuse^2 = opposite^2 + adjacent^2
Since the radius of the unit circle is 1, the hypotenuse of the triangle is 1. The opposite side is sin x, and the adjacent side is cos x. So, we get:
1^2 = sin^2x + cos^2x
which simplifies to:
sin^2x + cos^2x = 1
Hence, we have proven the identity sin^2x + cos^2x = 1 using the Pythagorean theorem and the geometry of the unit circle. This identity is extremely useful in trigonometry, as it relates the sine and cosine functions to each other and to the constant value 1.
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