Proving ( sin(x) + cos(x) = 1 )

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Among its many intriguing equations and identities, the equation $( \sin(x) + \cos(x) = 1 )$ is a fascinating and perplexing one. This blog post delves into the details of this equation, providing insights and a proof of its validity.

Table of Contents

Understanding the Basics

Before diving into the proof, let’s review some fundamental trigonometric identities and properties:

Pythagorean Identity:
$(\sin^2(x) + \cos^2(x) = 1)$

Angle Sum and Difference Identities:
$[\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)][\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)]$

Double-Angle Identities:
$[\sin(2x) = 2\sin(x)\cos(x)][\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1]$

Investigating the Equation $( \sin(x) + \cos(x) = 1 )$

To prove this equation, we will leverage the Pythagorean identity. Let’s start by squaring both sides of the equation:

(\sin(x) + \cos(x))^2 = 1^2
$]$

Expanding the left-hand side, we get:

$[\sin^2(x) + 2\sin(x)\cos(x) + \cos^2(x) = 1]$

Now, recall the Pythagorean identity $( \sin^2(x) + \cos^2(x) = 1 )$ . Substituting this into the equation, we have:

$[1 + 2\sin(x)\cos(x) = 1]$

Subtracting 1 from both sides, we obtain:

$[2\sin(x)\cos(x) = 0]$

Dividing both sides by 2, we get:

$[\sin(x)\cos(x) = 0]$

This implies that either $( \sin(x) = 0 )$ or $( \cos(x) = 0 )$ . Let’s consider these cases:

Case 1: $( \sin(x) = 0 )$ If $( \sin(x) = 0 )$ , then $( \cos(x) )$ must be 1 to satisfy the original equation $( \sin(x) + \cos(x) = 1 )$ . This occurs at angles $( x = 0, \pi, 2\pi, \ldots )$ .

Case 2: $( \cos(x) = 0 )$ If $( \cos(x) = 0 )$ , then $( \sin(x) )$ must be 1 to satisfy the original equation $( \sin(x) + \cos(x) = 1 )$ . This occurs at angles $( x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots )$ .

However, note that for $( \sin(x) )$ and $( \cos(x) )$ to be both 0 or both 1 at the same time is impossible due to their respective ranges and periodic nature.

Graphical Verification

For a more intuitive understanding, let’s visualize the functions $( \sin(x) )$ and $( \cos(x) )$ on a graph. Plotting these functions, we can observe their behavior and intersection points. The sum of $( \sin(x) )$ and $( \cos(x) )$ equals 1 only at specific points within their periodic cycles.

As shown in the graph, $( \sin(x) + \cos(x) )$ touches 1 at certain points, confirming our cases above.

Conclusion

The equation $( \sin(x) + \cos(x) = 1 )$ holds true only at specific angles where the trigonometric functions align to satisfy the sum. By understanding the fundamental identities and visualizing the functions graphically, we have demonstrated the validity of this intriguing trigonometric equation. While it might seem simple at first glance, it reveals the deep interconnectedness of trigonometric properties and identities.

Next time you encounter a trigonometric equation, remember that behind every sine and cosine function lies a world of mathematical beauty waiting to be explored.

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