Problem 34 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\cos (-x)-\sin (-x)=\cos x+\sin x$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\cos(x) + \sin(x)\).

Step 1: Understanding the Negative Angle Identities

Recall the trigonometric identities for negative angles: for cosine, \(\cos(-x) = \cos(x)\), and for sine, \(\sin(-x) = -\sin(x)\). These identities will help simplify the given expression.

Step 2: Apply Negative Angle Identities

Apply the identities: Replace \(\cos(-x)\) with \(\cos(x)\) and \(\sin(-x)\) with \(-\sin(x)\). The left-hand side of the equation becomes \(\cos(x) - (-\sin(x))\).

Step 3: Simplify the Expression

Simplify the expression \(\cos(x) - (-\sin(x))\). This is equivalent to \(\cos(x) + \sin(x)\).

Step 4: Verify Both Sides are Equal

Our simplification \(\cos(x) + \sin(x)\) matches the right side of the equation \(\cos x + \sin x\). Therefore, both sides of the identity are equal, verifying the given identity.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, denoted as \( \cos(x) \). It can be visualized on the unit circle where it represents the x-coordinate of a point rotating around the circle.
The cosine is an even function, which means it shows symmetry about the y-axis. This symmetry gives us the identity \( \cos(-x) = \cos(x) \), meaning that the cosine of a negative angle is the same as the cosine of the positive angle.
This property is very useful when simplifying trigonometric expressions, especially when verifying identities involving negative angles.

Sine Function

The sine function, represented by \( \sin(x) \), is another primary trigonometric function. It can be visualized on the unit circle as the y-coordinate of a point as it rotates around the circle.
Unlike the cosine, the sine function is an odd function. An odd function is symmetrical about the origin, and its defining identity is \( \sin(-x) = -\sin(x) \). This means if you take the sine of a negative angle, it's the negative of the sine of the positive angle.
Understanding this property is crucial for simplifying expressions and verifying trigonometric identities.

Negative Angles

Negative angles are defined as angles measured in the clockwise direction from the positive x-axis. In trigonometry, negative angles are often encountered, and their properties help simplify expressions.
The key trigonometric identities for negative angles are:

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

These identities enable us to manipulate and simplify equations by transforming negative angles into their positive counterparts, which are often easier to work with.

Verifying Identities

Verifying trigonometric identities involves proving that two sides of an equation are equal for all values for which both sides are defined. When tasked with verifying an identity, follow these steps:

• Identify and apply the appropriate trigonometric identities. For negative angles, use \( \cos(-x) = \cos(x) \) and \( \sin(-x) = -\sin(x) \).
• Simplify one or both sides of the equation using algebraic manipulation.
• Demonstrate that both sides of the equation simplify to the same expression.

In our example, after applying the negative angle identities, the expression \( \cos(-x) - \sin(-x) \) simplified to \( \cos(x) + \sin(x) \), which verifies the original identity.