Proving the Identity (sin^2 x cos^2 x 1) Using Trigonometric Definitions and the Pythagorean Theorem

Understanding and proving mathematical identities is crucial in advanced mathematics. In this article, we will explore and verify the identity (sin^2 x cos^2 x 1) using the definitions of sine and cosine in terms of right triangles and the Pythagorean theorem. Let’s dive into the details.

Using Trigonometric Definitions in Right Triangles

In a right triangle, the sine and cosine of an angle can be defined based on the side lengths. Consider a right triangle (ABC) where angle (B) is the right angle. The side opposite angle (B) is the hypotenuse, denoted as (h). The side adjacent to (B) is the base (b), and the side opposite to (B) is the perpendicular (p). By the Pythagorean theorem:

$$h^2 p^2 b^2$$

Proving (sin^2 x cos^2 x 1)

Let's define the trigonometric ratios for angle (x) in terms of the sides of the triangle:

$$sin x frac{p}{h}, quad cos x frac{b}{h}$$

Now we will square both sine and cosine and multiply them to see if the identity holds:

$$sin^2 x left(frac{p}{h}right)^2 frac{p^2}{h^2}$$

$$cos^2 x left(frac{b}{h}right)^2 frac{b^2}{h^2}$$

Now multiply these two results together:

$$sin^2 x cos^2 x frac{p^2}{h^2} times frac{b^2}{h^2} frac{p^2 b^2}{h^4}$$

Recall from the Pythagorean theorem that (p^2 b^2 h^2). Using this, we get:

$$p^2 b^2 (p^2 b^2)p^2 (h^2)p^2 - p^4 h^2p^2 - p^4$$

Since the terms cancel out, we are left with:

$$sin^2 x cos^2 x frac{p^2 b^2}{h^4} frac{(p^2 b^2)p^2 - p^4}{h^4} frac{h^4 - p^4}{h^4} frac{h^4}{h^4} 1$$

Using the Pythagorean Theorem Directly

Another way to prove this identity is by using the Pythagorean theorem directly. Consider a right triangle with sides (a, b, c), where (c) is the hypotenuse:

$$a^2 b^2 c^2$$

By definition of sine and cosine in a right triangle:

$$sin x frac{a}{c}, quad cos x frac{b}{c}$$

Now square both sine and cosine and multiply them:

$$sin^2 x left(frac{a}{c}right)^2 frac{a^2}{c^2}$$

$$cos^2 x left(frac{b}{c}right)^2 frac{b^2}{c^2}$$

Multiplying these two results together:

$$sin^2 x cos^2 x frac{a^2}{c^2} times frac{b^2}{c^2} frac{a^2 b^2}{c^4}$$

Using the Pythagorean theorem:

$$a^2 b^2 c^2$$

we can simplify the multiplication:

$$sin^2 x cos^2 x frac{(a^2 b^2)a^2 - a^4}{c^4} frac{c^2a^2 - a^4}{c^4} frac{c^4 - a^4}{c^4} frac{c^4}{c^4} 1$$

Conclusion

The identity (sin^2 x cos^2 x 1) holds true when we define sine and cosine in terms of the sides of a right triangle and apply the Pythagorean theorem. This proof shows the intimate relationship between trigonometric functions and the Pythagorean theorem, highlighting the beauty and consistency in mathematics.