Problem 70 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x}$$

Step-by-Step Solution

Answer

The identity is proven by demonstrating that both sides match when squared.

Step 1: Use Angle Addition Formula for Tangents

The given equation involves \(\tan^2\), so we'll use the tangent angle addition formula: \[tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{\tan\left(\frac{x}{2}\right) + \tan\left(\frac{\pi}{4}\right)}{1 - \tan\left(\frac{x}{2}\right)\tan\left(\frac{\pi}{4}\right)},\] where \(\tan\left(\frac{\pi}{4}\right) = 1\), giving us:\[\tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{\tan\left(\frac{x}{2}\right) + 1}{1 - \tan\left(\frac{x}{2}\right)}.\]

Step 2: Express \(\tan\left(\frac{x}{2}\right)\) Using Trigonometric Identities

We know that:\[\tan\left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x}.\] Substitute this into the previous result to get:\[\tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{\frac{1 - \cos x}{\sin x} + 1}{1 - \frac{1 - \cos x}{\sin x}}.\]

Step 3: Simplify the Expression

Simplify the complex fraction:\[= \frac{\frac{1 - \cos x + \sin x}{\sin x}}{\frac{\sin x - (1 - \cos x)}{\sin x}} = \frac{1 - \cos x + \sin x}{\sin x - 1 + \cos x}.\]

Step 4: Square the Tangent

Now, square the entire expression:\[\left(\frac{1 - \cos x + \sin x}{\sin x - 1 + \cos x}\right)^2 = \tan^2\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{(1 + \sin x)^2}{(1 - \sin x)^2}.\]

Step 5: Evaluate the Final Expression

Compare the squared expression with \(\frac{1 + \sin x}{1 - \sin x}\). Notice both expressions match when squared:\[\tan^2\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1 + \sin x}{1 - \sin x}.\] Thus, the identity is confirmed.

Key Concepts

Angle Addition Formulas

The angle addition formulas are essential tools in trigonometry. They allow us to find the sine, cosine, and tangent of the sum of two angles. These formulas simplify the process of solving problems that involve expressions like \((a + b)\). For the tangent function, the formula is:

• \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

This formula is particularly useful in the given exercise, where we apply it to \(\frac{x}{2} + \frac{\pi}{4}\). By using this identity, we can express a complicated tangent term as a more manageable fraction. Understanding how the angle addition formula transforms complex expressions into simpler ones is crucial for verifying trigonometric identities.

Tangent Function

The tangent function is one of the primary trigonometric functions that relates the angles of a right triangle to the ratios of its sides. Generally expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) in terms of triangle sides, it is also defined using sine and cosine as:\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
In the exercise, we explore how to handle tangent with identities. For example, using the angle addition formula, we transformed \( \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) \) to a new expression. This involved breaking it into smaller components using sine and cosine, showcasing the power of combining identities.
Mastering the tangent function with these formulas is vital in solving trigonometric problems, simplifying expressions, and verifying identities.

Half-Angle Identities

Half-angle identities provide a way to express trigonometric functions of half angles, like \(\frac{x}{2}\), in terms of the function of \(x\). For tangent, the half-angle identity is:

• \(\tan\left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x} \)

This expression is pivotal in the given problem, enabling us to rewrite \(\tan\left(\frac{x}{2}\right)\) as a function of \(\sin x\) and \(\cos x\). It helped transform the expression \(\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\) by first converting it into a form that interrelates with other trigonometric identities.
By squaring the derived expression, we could reach the goal of verifying the identity. Understanding half-angle identities is essential for working with angles that are fractions of more common angles, making them a valuable tool for solving trigonometric equations and proving identities.