Problem 78 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x $$

Step-by-Step Solution

Answer

The identity is verified; both sides simplify to \(4 \tan x \sec x\).

Step 1: Combine the fractions on the left side

The left side of the identity is \( \frac{1+\sin x}{1-\sin x} - \frac{1-\sin x}{1+\sin x} \). To combine these fractions, find a common denominator, which is \((1-\sin x)(1+\sin x) = 1-\sin^2 x\). Thus, the expression becomes: \( \frac{(1+\sin x)^2 - (1-\sin x)^2}{1-\sin^2 x} \).

Step 2: Simplify the expression on the numerator

Expand the squares in the numerator: \((1+\sin x)^2 = 1 + 2\sin x + \sin^2 x\) and \((1-\sin x)^2 = 1 - 2\sin x + \sin^2 x\). Subtracting these gives: \(1 + 2\sin x + \sin^2 x - 1 + 2\sin x - \sin^2 x = 4\sin x\).

Step 3: Simplify using the common denominator

Now, substitute the simplified numerator back, giving \( \frac{4\sin x}{1-\sin^2 x} \). Recall that \(1-\sin^2 x = \cos^2 x\), so this becomes \( \frac{4\sin x}{\cos^2 x} \).

Step 4: Convert the expression to trigonometric identities

Rewrite \( \frac{4\sin x}{\cos^2 x} \) using trigonometric identities. Since \( \frac{\sin x}{\cos x} = \tan x \) and \( \frac{1}{\cos x} = \sec x \), it follows that \( \frac{4\sin x}{\cos^2 x} = 4 \left( \frac{\sin x}{\cos x} \right) \left( \frac{1}{\cos x} \right) = 4 \tan x \sec x \).

Step 5: Verify the identity

Both sides of the equation are now \(4 \tan x \sec x\). This confirms that the original identity \( \frac{1+\sin x}{1-\sin x} - \frac{1-\sin x}{1+\sin x} = 4 \tan x \sec x \) is verified.

Key Concepts

Sine and Cosine

In trigonometry, the sine (\(\sin x\)) and cosine (\(\cos x\)) functions are fundamental in studying angles and their relationships. Let's simplify what these functions actually mean:

• The sine of an angle in a right triangle refers to the ratio of the opposite side to the hypotenuse. Mathematically, this is expressed as \(\sin x = \frac{\text{opposite}}{\text{hypotenuse}}\).
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse, or \(\cos x = \frac{\text{adjacent}}{\text{hypotenuse}}\).

These ratios are crucial in solving identity problems and are at the heart of our exercise. Notably, the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), is a helpful tool for simplification. In our exercise, this identity was used to transform \(1 - \sin^2 x\) into \(\cos^2 x\). This transformation helped simplify the given expression, showcasing the power of understanding sine and cosine relationships.

Algebraic Manipulation

Algebraic manipulation turns complex expressions into solvable equations. This involves finding common denominators, expanding expressions, or factoring. In the given exercise, we first focused on simplifying the equation by:

• Finding a common denominator: Each term in the original problem had different denominators, \(1-\sin x\) and \(1+\sin x\). We multiplied them to create a common denominator \((1-\sin x)(1+\sin x) = 1-\sin^2 x\), which simplified further into \(\cos^2 x\) when using the Pythagorean identity.
• Simplifying the numerator: The expression \((1+\sin x)^2 - (1-\sin x)^2\) was expanded to \(4\sin x\) by careful expansion and subtraction, illustrating an elegant example of simplifying algebraic expressions.

Understanding these steps is essential to manipulating expressions in trigonometric identities accurately.

Simplifying Fractions

Simplifying fractions often boils down to making them easier to understand. When dealing with trigonometric identities, simple fractions play a key role. Let's look at how this process works in our provided example:

• After simplifying the numerator and recognizing the common denominator as \(\cos^2 x\), we expressed the identity as \(\frac{4\sin x}{\cos^2 x}\).
• To further simplify, we recognized standard trigonometric identities: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). Using these, \(\frac{4\sin x}{\cos^2 x}\) becomes \(4 \tan x \sec x\).

Through these steps, fractions are not just simplified arithmetically but are aligned with trigonometric identities to clarify their true meaning and relationships in trigonometry.