Problem 78 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x $$

Step-by-Step Solution

Answer

Yes, the identity is verified.

Step 1: Simplify the Left Side

Start by finding a common denominator for the fractions \( \frac{1}{1-\sin x} - \frac{1}{1+\sin x} \). The common denominator is \((1-\sin x)(1+\sin x)\). Apply this to make a single fraction:\[\frac{(1+\sin x) - (1-\sin x)}{(1-\sin x)(1+\sin x)}\]

Step 2: Simplify the Numerator

Now simplify the numerator by distributing and combining like terms:\[1 + \sin x - 1 + \sin x = 2 \sin x\]

Step 3: Use the Pythagorean Identity in the Denominator

Recognize that the denominator \((1-\sin x)(1+\sin x)\) is a difference of squares, which simplifies to:\[1 - (\sin x)^2 = \cos^2 x\]

Step 4: Substitute and Further Simplify

Substitute the simplified numerator and the identity from Step 3 back into the fraction:\[\frac{2 \sin x}{\cos^2 x} = \frac{2 \sin x}{\cos x} \cdot \frac{1}{\cos x} = 2 \tan x \sec x\]

Step 5: Verify the Identity

Both sides of the equation are now \(2 \tan x \sec x\), thus verifying the identity:\[\frac{1}{1-\sin x} - \frac{1}{1+\sin x} = 2 \sec x \tan x\]

Key Concepts

Secant Function

The secant function, commonly written as \( \sec x \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function. This means:

• \( \sec x = \frac{1}{\cos x} \)

Because the secant function is the reciprocal of cosine, it is undefined wherever cosine is zero. These points are specifically at odd multiples of \( \frac{\pi}{2} \). This makes understanding its domain important.

To visualize, imagine the unit circle: as the angle \( x \) increases, the cosine value determines the horizontal component of a point on the circle. The secant stretches this while maintaining its structure and properties. The secant can also be used in calculus and various applications, such as solving trigonometric equations.

Tangent Function

The tangent function is represented as \( \tan x \) and is another fundamental trigonometric function. It is defined as the ratio of the sine function to the cosine function:

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

This function is unique because it represents the slope of the line that intersects the terminal side of an angle in the unit circle. The tangent function is periodic, repeating every \( \pi \, ext{(or 180 degrees)} \).

It is undefined wherever the cosine of the angle is zero because you cannot divide by zero! This occurs at angles such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \) (or 90°, 270°, etc.). For students learning trigonometry, understanding where the tangent is undefined is crucial, as these points will often appear in solving equations.

Pythagorean Identity

One of the most useful identities in trigonometry is the Pythagorean identity. It relates the square of the sine and cosine functions to form a fundamental equation:

• \( \sin^2 x + \cos^2 x = 1 \)

From this base identity, other forms can easily be derived. For instance:

• \( \tan^2 x + 1 = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

This identity is rooted in the Pythagorean theorem and is key in transforming and simplifying trigonometric expressions. It reveals that any change in the sine or cosine functions on the unit circle reflects in the other, maintaining balance. Whenever simplifying trigonometric identities, watch for opportunities to apply this powerful tool.

For example, in the given exercise, the identity was used to recognize and simplify a difference of squares, transforming it into a usable form for the solution.

Difference of Squares

The difference of squares is an algebraic concept where two squares are subtracted to form an expression:\[ a^2 - b^2 = (a - b)(a + b) \]

This idea can be applied in various mathematical problems, showing its versatility. It helps simplify expressions, especially useful in factoring and solving equations. In trigonometry, identifying a difference of squares can aid in simplifying expressions like:\[(1 - \sin x)(1 + \sin x) = 1 - \sin^2 x = \cos^2 x\]

In the given exercise, recognizing that \( (1 - \sin x)(1 + \sin x) \) is a difference of squares leads to a key simplification using the Pythagorean identity \( 1 - \sin^2 x = \cos^2 x \). This method is a neat trick to simplify calculations involving trigonometric identities, turning a seemingly complex fraction into a simple form compatible with other trigonometric expressions.