Trigonometry identities - 1

Trigonometric identity: sin²(θ) + cos²(θ) = 1

In order to prove this trigonometric identity, you will apply the Pythagorean Theorem. Let us consider a right angled triangle ABC as shown below:

Image 1: Proof of sin²(x) + cos²(x) =1

In the above right triangle, with respect to angle theta, the trigonometric ratios of sine and cosine are defined as follows:

Image 2: Proof of sin²(x) + cos²(x) =1

 By applying Pythagorean Theorem in the above triangle, we get the following result:

(AB)² + (BC)² = (AC)²

Notice that in the trigonometric ratios (1) and (2) defined above, we have AC in the denominator of both fractions. Thus, in order to include the trigonometric ratios in this equation, we will divide the equation by (AC)². The equation then becomes,

Image 3: Proof of sin²(x) + cos²(x) =1

In this equation, (AC)² over itself (on the right side) is equal to 1. From result (1) obtained above, we know that sin(θ) is AB over AC and cos(θ) is BC over AC. Applying this , we get

Image 4: Proof of sin²(x) + cos²(x) =1

Thus we arrive at a result, which is one of the most important trigonometric identities in the study of Trigonometry.