Problem 34 - Precalculus

Question

Prove or disprove the identity. $$\frac{1}{1+\cos x}-\frac{1}{1-\cos (-x)}=-2 \cot x \csc x$$

Step-by-Step Solution

Answer

The given identity is true.

Step 1: Simplify the Left Side

First, recognize that: \(\cos(-x) = \cos(x)\). Substitute this into the equation:\[\frac{1}{1 + \cos x} - \frac{1}{1 - \cos x}\]This simplifies to:\[\frac{1}{1 + \cos x} - \frac{1}{1 - \cos x}\]

Step 2: Find a Common Denominator

The common denominator for the fractions \(\frac{1}{1 + \cos x}\) and \(\frac{1}{1 - \cos x}\) is \((1 + \cos x)(1 - \cos x)\). Thus, rewrite the equation as:\[\frac{1}{1 + \cos x} = \frac{1 - \cos x}{(1 + \cos x)(1 - \cos x)}\]\[\frac{1}{1 - \cos x} = \frac{1 + \cos x}{(1 + \cos x)(1 - \cos x)}\]Subtract to get:\[\frac{1 - \cos x}{(1 + \cos x)(1 - \cos x)} - \frac{1 + \cos x}{(1 + \cos x)(1 - \cos x)}\]

Step 3: Simplify the Numerator

Notice that the denominator simplifies to \(1 - (\cos x)^2 = \sin^2 x\) using the Pythagorean identity. Simplify the numerator:\[(1 - \cos x) - (1 + \cos x) = -2\cos x\]So, the expression becomes:\[\frac{-2\cos x}{\sin^2 x}\]

Step 4: Express in Terms of Trigonometric Functions

Re-express \(\frac{-2\cos x}{\sin^2 x}\) using trigonometric identities:\[-2 \frac{\cos x}{\sin x} \frac{1}{\sin x} = -2 \cot x \csc x\]

Step 5: Compare Both Sides of the Identity

The simplified left side expression \(-2 \cot x \csc x\) matches the right side of the original identity. Therefore, the identity is proven to be true.

Key Concepts

Pythagorean identity

The Pythagorean identity is a non-negotiable truth in trigonometry that links the sine and cosine functions. Its basic format is given by the equation: \[\sin^2 x + \cos^2 x = 1\]This identity allows us to relate the squares of sine and cosine functions to the number one, acting as a vital bridge in many trigonometric proofs and simplifications.
When dealing with expressions involving trigonometric functions, this identity often helps simplify the denominator or transform the problem into a more solvable form.

• In our exercise, once the expression \((1 + \cos x)(1 - \cos x)\) appears in the denominator, we utilize the Pythagorean identity to simplify it to \(\sin^2 x\).
• This transformation not only simplifies the expression but also enables further manipulation of the equation using sine and cosine ratios.

Understanding and recognizing opportunities to apply the Pythagorean identity is fundamental for mastering trigonometric simplifications.

Trigonometric simplification

Trigonometric simplification is a process used to make trigonometric equations more manageable. This process relies on identities and known ratios to reduce or re-express complex terms more simply.
In our given exercise, simplification occurs after recognizing that our left-side expression of \(\frac{1}{1+\cos x} - \frac{1}{1-\cos x}\) can be combined using a common denominator. This leads us to:

• Combine fractions by finding a common denominator \((1 + \cos x)(1 - \cos x)\), which simplifies using the Pythagorean identity to \(\sin^2 x\).
• The resulting expression \(-2 \frac{\cos x}{\sin^2 x}\) is further transformed by factoring out common terms and employing trigonometric identities, specifically those known for cotangent and cosecant.

Trigonometric simplifications, like in this example, significantly clear up the problem, reducing clutter and moving steps closer to the desired form for easier computations.

Cotangent and cosecant functions

The cotangent (\(\cot x\)) and cosecant (\(\csc x\)) functions are reciprocal trigonometric functions. They are defined as follows:

• Cotangent is the reciprocal of tangent, i.e., \(\cot x = \frac{\cos x}{\sin x}\).
• Cosecant is the reciprocal of sine, i.e., \(\csc x = \frac{1}{\sin x}\).

These functions are inherently linked to the sine and cosine functions. By expressing terms in equations through cotangent and cosecant, the steps in trigonometric proofs become streamlined.
In the exercise's solution, we reached \(-2 \frac{\cos x}{\sin x} \frac{1}{\sin x}\), which translates to \(-2 \cot x \csc x\) using the definitions of cotangent and cosecant.
When resolving trigonometric identities, using these expressions can lead to a condensed and clearer final form, as seen in the proof, where the simplification perfectly matches the given identity.