Problem 65

Question

Verify each identity. $$ \cot \frac{x}{2}=\frac{\sin x}{1-\cos x} $$

Step-by-Step Solution

Verified

Answer

The provided identity is verified, $\cot \frac{x}{2}=\frac{\sin x}{1-\cos x}$, which simplifies to $\frac{\cos \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta}$.

1Step 1: Rewrite cotangent in terms of sine and cosine

Rewrite $\cot \frac{x}{2}$ as $\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}$. Therefore, the original equation becomes $\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} = \frac{\sin x}{1-\cos x}$.

2Step 2: Apply the double-angle formulas

Apply the double-angle formulas $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos{2\theta} = 1 - 2\sin^2 \theta$. In $\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} = \frac{\sin x}{1-\cos x}$, substitute $x$ for $2\theta$ and solve to get $\frac{\cos \theta}{\sin \theta} = \frac{2\sin\theta\cos\theta}{1- (1 - 2\sin^2 \theta)}$.

3Step 3: Simplify the equation

Simplify the equation $\frac{\cos \theta}{\sin \theta} = \frac{2\sin\theta\cos\theta}{2\sin^2 \theta}$ by eliminating common factors. The equation simplifies to $\frac{\cos \theta}{\sin\theta} = \frac{\cos \theta}{\sin \theta}$.

Key Concepts

Double-Angle FormulasCotangent IdentitySine and Cosine Relationships

Double-Angle Formulas

The double-angle formulas help us express trigonometric functions of double angles in terms of single angles. They are particularly useful when solving trigonometric identities or equations. Here, we use the double-angle formulas equivalent for sine and cosine, which are:

$ \sin 2\theta = 2 \sin \theta \cos \theta $
$ \cos 2\theta = 1 - 2\sin^2 \theta $

These formulas help us convert expressions involving $ x $ into expressions involving $ \theta $ or $ \frac{x}{2} $, facilitating simplification. In our problem, we substitute $ x $ with $ 2\theta $ which allows the expression to be rearranged and factors simplified, ultimately showing both sides of the identity match.

Cotangent Identity

Cotangent, often abbreviated as "cot," is one of the basic trigonometric functions and is the reciprocal of tangent. The identity we are dealing with connects cotangent to sine and cosine in a specific way. We started with the identity:

$ \cot \frac{x}{2} = \frac{\sin x}{1 - \cos x} $

Rewriting cotangent in terms of sine and cosine helps make the given identity more manageable.

To rewrite $ \cot \frac{x}{2} $, we use:

$ \cot \theta = \frac{\cos \theta}{\sin \theta} $

This transformation is crucial in allowing us to express both sides of the equation in a similar form, making it easier to verify the identity.

Sine and Cosine Relationships

Sine and cosine functions have a strong relationship that is often utilized in trigonometry to simplify and solve equations. Recognizing these relationships is key when working with identities, as seen in the problem above.Here are a few important points about sine and cosine:

Sine and cosine are periodic with respect to the angle, meaning they repeat their values in cycles over specified intervals.
Their fundamental relationship is given by $ \sin^2 \theta + \cos^2 \theta = 1 $.

In our verification problem, understanding these relationships allows us to manipulate the original identity into a form where the same functions appear on both sides, confirming that the identity is indeed true. By simplifying expressions using trigonometric properties, we uncover useful insights, like common factors that can be eliminated, simplifying expressions further.

Problem 65

Problem 66

Other exercises in this chapter

Problem 65

Use an identity to solve each equation on the interval $[0,2 \pi)$ $$ \sin ^{2} x-2 \cos x-2=0 $$

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Problem 65

Find all zeros of $f(x)=x^{3}-2 x^{2}-5 x+6$

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Problem 66

Use an identity to solve each equation on the interval $[0,2 \pi)$ $$ 4 \sin ^{2} x+4 \cos x-5=0 $$

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Problem 66

will help you prepare for the material covered in the next section. $$ \text { Solve: } 2\left(1-u^{2}\right)+3 u=0 $$

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