Discover The Pythagorean Identity: Sin^2X + Cos^2X = 1

sin^2x + cos^2x =

1

The expression sin^2x + cos^2x equals 1.

This is a fundamental trigonometric identity known as the Pythagorean identity. It states that the sum of the squares of the sine and cosine of an angle is equal to 1.

To see why this is true, we can use the fact that sinx and cosx are related to the sides of a right triangle. Let’s consider a right triangle with hypotenuse of length 1, an angle x opposite to one of the legs and the second angle opposite to the other leg of the right triangle. The sine of x is defined as the length of the opposite leg divided by the length of the hypotenuse. The cosine of x is defined as the length of the adjacent leg divided by the length of the hypotenuse.

So we have sinx = opposite / hypotenuse and cosx = adjacent / hypotenuse. Squaring both of these equations, we get:

sin^2x = (opposite / hypotenuse)^2 = opposite^2 / hypotenuse^2

cos^2x = (adjacent / hypotenuse)^2 = adjacent^2 / hypotenuse^2

Adding these two equations, we get:

sin^2x + cos^2x = (opposite^2 + adjacent^2) / hypotenuse^2

Since the opposite and adjacent sides are the legs of the right triangle and they are perpendicular, we can apply the Pythagorean theorem and get that opposite^2 + adjacent^2 = hypotenuse^2.

Substituting this into the expression above, we get:

sin^2x + cos^2x = hypotenuse^2 / hypotenuse^2 = 1

Therefore, sin^2x + cos^2x equals 1, as we set out to show.

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