Question

Expand and simplify the expression \((\sec x-\tan x)(\sin x+1)\) to a primary trigonometric function.

Short answer

\( (\text{sec } x - \text{tan } x)(\text{sin } x + 1) = \text{cos } x \)

Step-by-step solution

- Distribute Each Term

Start by using the distributive property (also known as the FOIL method) to expand the expression \((\text{sec } x - \text{tan } x)(\text{sin } x + 1)\). We distribute each term in the first parenthesis to each term in the second parenthesis: \[ (\text{sec } x)(\text{sin } x) + (\text{sec } x)(1) - (\text{tan } x)(\text{sin } x) - (\text{tan } x)(1) \]

- Use Trigonometric Identities

Next, apply trigonometric identities to simplify each term. Recall the identities: \[\text{sec } x = \frac{1}{\text{cos } x} \] and \[\text{tan } x = \frac{\text{sin } x}{\text{cos } x} \]. Substitute these identities into the expression: \[ \left(\frac{1}{\text{cos } x}\right)(\text{sin } x) + \left(\frac{1}{\text{cos } x}\right)(1) - \left(\frac{\text{sin } x}{\text{cos } x}\right)(\text{sin } x) - \left(\frac{\text{sin } x}{\text{cos } x}\right)(1) \]

- Simplify Each Term

Simplify each term: \[ \frac{\text{sin } x}{\text{cos } x} + \frac{1}{\text{cos } x} - \frac{\text{sin}^2 x}{\text{cos } x} - \frac{\text{sin } x}{\text{cos } x} \] Combine like terms: \[ \left(\frac{\text{sin } x - \text{sin } x}{\text{cos } x}\right) + \left(\frac{1 - \text{sin}^2 x}{\text{cos } x} \right) = \frac{1 - \text{sin}^2 x}{\text{cos } x} \]

- Apply Pythagorean Identity

Use the Pythagorean identity \[1 - \text{sin}^2 x = \text{cos}^2 x \] to simplify the expression further: \[ \frac{\text{cos}^2 x}{\text{cos } x} = \text{cos } x \]

Key concepts

Distributive Property

When faced with the expression \( (\text{sec } x - \text{tan } x)(\text{sin } x + 1) \), the first step is to expand it using the distributive property. The distributive property helps us multiply each term inside the parenthesis by each term in the other parenthesis. Using this property, we get:

\[ (\text{sec } x)(\text{sin } x) + (\text{sec } x)(1) - (\text{tan } x)(\text{sin } x) - (\text{tan } x)(1) \]

This method ensures we don't miss any terms while expanding the expression.

Trigonometric Simplification

After distributing, we simplify the trigonometric expressions. Use the basic trigonometric identities:

• \(\text{sec } x = \frac{1}{\text{cos } x} \)
• \(\text{tan } x = \frac{\text{sin } x}{\text{cos } x} \)

Substitute these into our expanded expression:

\[ \frac{\text{sin } x}{\text{cos } x} + \frac{1}{\text{cos } x} - \frac{\text{sin}^2 x}{\text{cos } x} - \frac{\text{sin } x}{\text{cos } x} \]

This step transforms the trigonometric terms into familiar fractions.

Pythagorean Identity

To continue simplifying, we use the Pythagorean identity, which states that:

\[ 1 - \text{sin}^2 x = \text{cos}^2 x \]

Substitute this identity into the expression we derived:

\[ \frac{1 - \text{sin}^2 x}{\text{cos } x} = \frac{\text{cos}^2 x}{\text{cos } x} \]

This application greatly simplifies the expression.

Secant Function

The secant function, denoted as \( \text{sec } x \), is the reciprocal of the cosine function. This means:

\[ \text{sec } x = \frac{1}{\text{cos } x} \]

We used this identity to replace secant in our equation to better work with the trigonometric simplifications. It's a key step whenever you encounter secant in expressions.

Tangent Function

Similarly, the tangent function, denoted as \( \text{tan } x \), is the ratio of the sine and cosine functions:

\[ \text{tan } x = \frac{\text{sin } x}{\text{cos } x} \]

This identity allows us to break down and simplify expressions involving tangent. Just like with secant, understanding and applying the basic trigonometric identities makes complex expressions easier to handle.