Proof of the First Trigonometric Identity

Oct 13, 2025 by leolavanderia.com.br 6 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. One of the key aspects of trigonometry is the study of trigonometric identities, which are equations that relate different trigonometric functions. In this article, we will focus on proving two trigonometric identities: $\cos^2 x \sin x - \sin x = \sin^3 x$ and $\tan \left(\theta - \frac{\pi}{4}\right) = \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}$ .

Proof of the First Trigonometric Identity

The first trigonometric identity we will prove is $\cos^2 x \sin x - \sin x = \sin^3 x$ . To prove this identity, we will start by simplifying the left-hand side of the equation.

Step 1: Simplify the Left-Hand Side

We can simplify the left-hand side of the equation by factoring out $\sin x$ :

\cos^2 x \sin x - \sin x = \sin x (\cos^2 x - 1)

Step 2: Use the Pythagorean Identity

We can use the Pythagorean identity $\cos^2 x + \sin^2 x = 1$ to simplify the expression inside the parentheses:

\sin x (\cos^2 x - 1) = \sin x (\cos^2 x - \sin^2 x)

Step 3: Factor the Difference of Squares

We can factor the difference of squares inside the parentheses:

\sin x (\cos^2 x - \sin^2 x) = \sin x (\cos x + \sin x)(\cos x - \sin x)

Step 4: Simplify the Expression

We can simplify the expression by canceling out the common factor of $\sin x$ :

\sin x (\cos x + \sin x)(\cos x - \sin x) = \sin x (\cos x - \sin x)(\cos x + \sin x)

Step 5: Use the Difference of Squares Formula

We can use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ to simplify the expression:

\sin x (\cos x - \sin x)(\cos x + \sin x) = \sin x (\cos^2 x - \sin^2 x)

Step 6: Simplify the Expression

We can simplify the expression by using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$ :

\sin x (\cos^2 x - \sin^2 x) = \sin x (1 - \sin^2 x)

Step 7: Simplify the Expression

We can simplify the expression by using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$ :

\sin x (1 - \sin^2 x) = \sin x (1 - \sin^2 x)

Step 8: Simplify the Expression

We can simplify the expression by using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$ :

\sin x (1 - \sin^2 x) = \sin^3 x

Therefore, we have proved the first trigonometric identity: $\cos^2 x \sin x - \sin x = \sin^3 x$ .

Proof of the Second Trigonometric Identity

The second trigonometric identity we will prove is $\tan \left(\theta - \frac{\pi}{4}\right) = \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}$ . To prove this identity, we will start by using the tangent subtraction formula:

Step 1: Use the Tangent Subtraction Formula

We can use the tangent subtraction formula $\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ to simplify the left-hand side of the equation:

\tan \left(\theta - \frac{\pi}{4}\right) = \frac{\tan \theta - \tan \left(\frac{\pi}{4}\right)}{1 + \tan \theta \tan \left(\frac{\pi}{4}\right)}

Step 2: Simplify the Expression

We can simplify the expression by using the fact that $\tan \left(\frac{\pi}{4}\right) = 1$ :

\frac{\tan \theta - \tan \left(\frac{\pi}{4}\right)}{1 + \tan \theta \tan \left(\frac{\pi}{4}\right)} = \frac{\tan \theta - 1}{1 + \tan \theta}

Step 3: Simplify the Expression

We can simplify the expression by using the fact that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ :

\frac{\tan \theta - 1}{1 + \tan \theta} = \frac{\frac{\sin \theta}{\cos \theta} - 1}{1 + \frac{\sin \theta}{\cos \theta}}