Problem 18 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{\sin x}{\csc x}+\frac{\cos x}{\sec x} $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Convert Cosecant and Secant

First, rewrite the reciprocals in terms of sine and cosine. Recall that \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). Substitute these into the original expression to get: \[ \frac{\sin x}{\frac{1}{\sin x}} + \frac{\cos x}{\frac{1}{\cos x}} \]

Step 2: Simplify Each Fraction

Simplify each fraction by multiplying by the reciprocal: \[ \sin x \cdot \sin x + \cos x \cdot \cos x \] This simplifies to \( \sin^2 x + \cos^2 x \).

Step 3: Use Pythagorean Identity

Apply the Pythagorean identity, which states that \( \sin^2 x + \cos^2 x = 1 \). Substitute this identity into the expression to simplify it to \( 1 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are key tools for simplifying trigonometric expressions and solving trigonometric equations. Some of the fundamental identities include:

• Pythagorean identities: These express relationships between sine, cosine, and tangent functions.
• Reciprocal identities: These relate the basic trigonometric functions to each other, such as how cosecant is the reciprocal of sine.
• Quotient identities: These involve the quotient of two trigonometric functions, such as tangent and cotangent.

In exercises like the one given where you simplify \( rac{\sin x}{\csc x}+\frac{\cos x}{\sec x} \), these identities are crucial. They help in converting the expression into equivalent forms, making it easier to simplify.

Cosecant and Secant

Cosecant and secant are functions associated with the basic trigonometric ratios specifically derived from the sine and cosine functions. Understanding their reciprocal nature is vital for trigonometric simplification:

• Cosecant (\(\csc x\)): As the reciprocal of sine, cosecant is defined as \( \csc x = \frac{1}{\sin x} \). This means, whenever you encounter \(\csc x\), you can substitute \(\frac{1}{\sin x}\) to simplify equations.
• Secant (\(\sec x\)): Similarly, secant is the reciprocal of cosine, defined as \( \sec x = \frac{1}{\cos x} \). By recognizing and using these relationships, you can transform complex expressions into simpler fractions involving sine and cosine.

Applying these definitions helps transform the original expression into something that can be easily simplified by multiplication, eventually leading to utilizing known identities.

Pythagorean Identity

Among the cornerstone trigonometric identities, the Pythagorean identity plays a pivotal role. It states that for any angle \(x\), the sum of the square of sine and the square of cosine always equals one:\[ \sin^2 x + \cos^2 x = 1 \]This identity is essential in converting expressions like \( \sin^2 x + \cos^2 x \) into the simpler form of \(1\). It's derived from the Pythagorean theorem, hence the name, and applies directly to the unit circle.In the exercise, after simplifying the expression \( \sin x \cdot \sin x + \cos x \cdot \cos x \), you recognize this critical identity. By substituting the identity, the expression simplifies to \(1\), showcasing the power and usefulness of the Pythagorean identity in trigonometric simplification.