Problem 38 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1 $$

Step-by-Step Solution

Answer

Verified: the expression equals 1 using trigonometric identities.

Step 1: Rewrite Reciprocal Trigonometric Functions

Rewrite the reciprocal trigonometric functions in terms of cosine and sine. Recall that \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). This allows us to rewrite the fractions: \[ \frac{\cos x}{\sec x} = \cos x \cdot \cos x = \cos^2 x \] \[ \frac{\sin x}{\csc x} = \sin x \cdot \sin x = \sin^2 x \].

Step 2: Combine like terms

Add the expressions to simplify the left side of the equation. Since they are now in terms of \( \cos^2 x \) and \( \sin^2 x \), you can simply add them together:\[ \cos^2 x + \sin^2 x \]

Step 3: Apply Pythagorean Identity

Recall the fundamental Pythagorean identity: \[ \cos^2 x + \sin^2 x = 1 \].Substitute this into your expression:\[ \cos^2 x + \sin^2 x = 1 \].

Key Concepts

Reciprocal Trigonometric Functions

Trigonometric functions are essential in understanding angles and lengths in triangles and other geometric shapes. Among these functions, there are what we call reciprocal trigonometric functions. These are derived by taking the reciprocal of the primary trigonometric functions. Let's look closer at sine and cosine and their reciprocals:

\(\sec x\) and \(\csc x\) are the reciprocal trigonometric functions for cosine and sine respectively.

• The secant (\(\sec x\)) is the reciprocal of the cosine function: \(\sec x = \frac{1}{\cos x}\).
• The cosecant (\(\csc x\)) is the reciprocal of the sine function: \(\csc x = \frac{1}{\sin x}\).

To solve trigonometric identities involving these functions, it's often useful to rewrite them in terms of sine and cosine. This transforms \(\frac{\cos x}{\sec x}\) into \(\cos x \cdot \cos x = \cos^2 x\), and \(\frac{\sin x}{\csc x}\) into \(\sin x \cdot \sin x = \sin^2 x\). Rewriting in these terms makes it easier to manipulate the expressions and apply additional trigonometric identities.

Pythagorean Identity

One of the cornerstone identities in trigonometry is the Pythagorean identity. It arises from the Pythagorean theorem and relates the sine and cosine functions.

The Pythagorean identity is written as: \[\cos^2 x + \sin^2 x = 1\]This identity highlights the fundamental relationship between squared cosine and sine values. It tells us that no matter what angle \(x\) is, the sum of \(\cos^2 x\) and \(\sin^2 x\) will always equal 1.

This identity becomes very handy when verifying other trigonometric identities by simplifying expressions with squares of sine and cosine. When we see \(\cos^2 x + \sin^2 x\), we can directly substitute it with 1 thanks to this identity. Therefore, in the given problem: substituting \(\cos^2 x + \sin^2 x\) with 1 confirms that the left side of the equation equals 1, thereby verifying the identity.

Verifying Identities

Verifying trigonometric identities is a methodical process where you prove the equality of both sides of an equation using known identities and algebraic manipulation. To verify an identity efficiently, follow these general steps:

• Substitute any known trigonometric identities (like the reciprocal or Pythagorean identities) into the equation.
• Simplify fractions or expressions into basic sine and cosine terms whenever possible.
• Combine like terms and simplify further until you reach an obvious equality like \(1 = 1\).

In the exercise given, we've rewritten the reciprocal functions \(\frac{\cos x}{\sec x}\) and \(\frac{\sin x}{\csc x}\) in terms of sine and cosine. We then combined them using the Pythagorean identity, simplifying directly to 1, which confirmed the trigonometric identity. This systematic approach ensures that identities are verified with clarity and precision.