sin^2x + cos^2x =
1
1
The identity: sin^2x + cos^2x = 1 is one of the most fundamental identities in trigonometry. This identity is also known as the Pythagorean identity.
Here’s a step-by-step explanation of how to prove the identity:
1. Start with the definition of the trigonometric functions:
sin(x) = opposite / hypotenuse, and cos(x) = adjacent / hypotenuse
2. Since we’re dealing with a right triangle, we can use the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse:
opposite^2 + adjacent^2 = hypotenuse^2
3. Divide each term of the equation by hypotenuse^2 to get:
opposite^2 / hypotenuse^2 + adjacent^2 / hypotenuse^2 = 1
4. Recall that sin(x) = opposite / hypotenuse and cos(x) = adjacent / hypotenuse, which we can substitute into the equation:
sin^2(x) + cos^2(x) = 1
5. Therefore, sin^2(x) + cos^2(x) = 1 is the Pythagorean identity, and it holds true for any angle x.
In summary, the identity sin^2(x) + cos^2(x) = 1 can be proved using basic trigonometry definitions and the Pythagorean theorem.
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