Understanding the Pythagorean Identity: Proving sin²x + cos²x = 1 using a Right Triangle and Trigonometric Ratios

sin²x + cos²x = 1

To prove the equation sin²x + cos²x = 1, we will use one of the fundamental trigonometric identities known as the Pythagorean Identity

To prove the equation sin²x + cos²x = 1, we will use one of the fundamental trigonometric identities known as the Pythagorean Identity.

The Pythagorean Identity states that for any angle x in a right triangle, the sum of the squares of the sine and cosine of x is always equal to 1.

To understand how to prove this identity, let’s begin by considering a right triangle:

We will label the sides of the right triangle as follows:

– The length of the hypotenuse will be represented by ‘c’.
– The length of the side adjacent to angle x will be represented by ‘a’.
– The length of the side opposite to angle x will be represented by ‘b’.

Based on the definitions above, the sine of angle x is defined as the ratio of the length of the side opposite to x (b) to the length of the hypotenuse (c), which can be written as sin(x) = b/c.

Similarly, the cosine of angle x is defined as the ratio of the length of the side adjacent to x (a) to the length of the hypotenuse (c), which can be written as cos(x) = a/c.

Using these definitions, we can find the square of the sine and cosine of x:

sin²x = (b/c)² = b²/c²
cos²x = (a/c)² = a²/c²

Now, let’s add these two equations together:

sin²x + cos²x = b²/c² + a²/c²

Notice that both terms share a common denominator, which is c². We can combine the numerators:

sin²x + cos²x = (b² + a²)/c²

Using the Pythagorean Theorem, we know that in a right triangle, the sum of the squares of the lengths of the two shorter sides (a and b) is equal to the square of the length of the hypotenuse (c) squared:

a² + b² = c²

Therefore, we can substitute a² + b² with c² in our equation:

sin²x + cos²x = (a² + b²)/c² = c²/c² = 1

Hence, we have proven that sin²x + cos²x = 1.

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