Problem 67 - Precalculus Essentials

Question

In Exercises \(59-68\), verify each identity. $$\cot \frac{x}{2}=\frac{1+\cos x}{\sin x}$$

Step-by-Step Solution

Answer

After substitifying cotangent in terms of sine and cosine, using half-angle formulas and simplifying both sides, it is observed that the left hand side is equal to the right hand side, thus verifying the given trigonometric identity.

Step 1: Write the given identity

We are given that \( \cot \frac{x}{2}=\frac{1+\cos x}{\sin x} \) and asked to verify it.

Step 2: Express cotangent in terms of sine and cosine

We know that \( \cot x = \frac{ \cos x }{ \sin x } \), so we can rewrite \( \cot \frac{x}{2} \) as \( \frac{ \cos \frac{x}{2} }{ \sin \frac{x}{2} } \). The identity is now \( \frac{ \cos \frac{x}{2} }{ \sin \frac{x}{2} } = \frac{1+\cos x}{\sin x} \)

Step 3: Use half-angle formulas

From the half-angle formulas, we know that \( \cos \frac{x}{2} = \sqrt{ \frac{1 + \cos x}{2} } \) and \( \sin \frac{x}{2} = \sqrt{ \frac{1 - \cos x}{2} } \). Substituting these into our identity we get \( \frac{ \sqrt{ \frac{1 + \cos x}{2} } }{ \sqrt{ \frac{1 - \cos x}{2} } } = \frac{1+\cos x}{\sin x} \)

Step 4: Simplify both sides

Solving the left hand side, we have \( \frac{ \sqrt{ \frac{1 + \cos x}{2} } }{ \sqrt{ \frac{1 - \cos x}{2} } } = \sqrt{ \frac{ \frac{1 + \cos x}{2} }{ \frac{1 - \cos x}{2} } } = \sqrt{ \frac{1+\cos x}{1-\cos x} } \). Now, we know that \( \frac{1}{\sin x} = \csc x \) and \( \csc x \) can be expressed as \( \sqrt{ \frac{1}{1 - \cos^2 x} } \), so the right hand side can be written as \( (1 + \cos x)\sqrt{ \frac{1}{1 - \cos^2 x} } = \sqrt{ \frac{1 + \cos x}{1 - \cos^2 x} } \)

Step 5: Check if LHS equals RHS

If we compare the left hand side (LHS) and the right hand side (RHS) of the equation after the simplification, we see that both sides are identical, thus verifying the identity.

Key Concepts

Cotangent

In trigonometry, the cotangent function is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. Mathematically, it is expressed as: \[\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}.\]

• This formula shows that cotangent is the ratio of the cosine and sine of an angle.
• When the angle is zero or a multiple of \(\pi\), special care is needed because the sine becomes zero, making cotangent undefined at these points.

The identity for the cotangent of a half angle is particularly useful. It allows us to relate the cotangent of half an angle directly to the sine and cosine of the full angle, simplifying complex expressions. Understanding this can also help solve identities and equations involving trigonometric functions.

Half-Angle Formulas

Half-angle formulas are special trigonometric identities that allow you to find the sine, cosine, or tangent of half of a given angle. These formulas are derived from the double-angle formulas and are very useful in trigonometric simplification and integration.
For cosine and sine, the half-angle formulas are:\[\cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}}\]\[\sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}}\]

• These formulas can be a bit tricky, as the square root introduces a plus or minus sign, which depends on the quadrant in which the angle \(\frac{x}{2}\) is located.
• They are particularly helpful in solving trigonometric identities or simplifying expressions involving angles like \(\frac{x}{2}\).

Using these half-angle identities allows us to transform more complex trigonometric equations into simpler forms, making them easier to understand and solve.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms. This can be achieved through identities, such as Pythagorean, sum and difference, and half-angle identities.

• Key to simplification is the ability to recognize patterns that match these known identities.
• Common techniques include combining terms, factoring, or canceling out terms using identities.

In our identity verification, \[\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} = \frac{1+\cos x}{\sin x} \]we employed trigonometric simplification techniques using half-angle formulas. By simplifying both sides of the equation, we found equivalences that support the equality. Such steps not only help in solving exercises but are fundamental in understanding deeper trigonometric properties and proofs.