Problem 44 - Algebra and Trigonometry

Question

Verify the identity. $$ \csc x-\sin x=\cos x \cot x $$

Step-by-Step Solution

Answer

The identity is verified: \( \csc x - \sin x = \cos x \cot x \).

Step 1: Write Down the Identity

We start with the equation we need to prove: \[ \csc x - \sin x = \cos x \cot x \] Our goal is to manipulate one or both sides of the equation to show they are equal.

Step 2: Express in Terms of Sine and Cosine

Rewrite each trigonometric function in terms of sine and cosine:\[ \csc x = \frac{1}{\sin x} \] \[ \cot x = \frac{\cos x}{\sin x} \] Substitute these identities into the equation.

Step 3: Substitute and Simplify Left Side

Replace \( \csc x \) with \( \frac{1}{\sin x} \) in the left side of the equation:\[ \frac{1}{\sin x} - \sin x \] Combine into a single fraction:\[ \frac{1 - \sin^2 x}{\sin x} \]

Step 4: Use Pythagorean Identity

Utilize the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to replace \(1 - \sin^2 x\) with \(\cos^2 x\):\[ \frac{\cos^2 x}{\sin x} \]

Step 5: Simplify and Compare

Factor out the \(\cos x\): \[ \cos x \cdot \frac{\cos x}{\sin x} \]Recognize that \( \frac{\cos x}{\sin x} \) is \( \cot x \), so the expression simplifies to:\[ \cos x \cot x \]

Step 6: Conclusion of Identity Verification

The simplified left side matches the right side \( \cos x \cot x \), thus verifying the identity holds true.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, expressing a core relationship between sine and cosine. It states that \[ \sin^2 x + \cos^2 x = 1 \]. This identity is derived from the Pythagorean theorem applied to a unit circle. Consider a right triangle where the hypotenuse is the radius of a unit circle:

• The sine of an angle is the length of the opposite side over the hypotenuse.
• The cosine of an angle is the length of the adjacent side over the hypotenuse.

Since the hypotenuse is 1 in the unit circle, the lengths of the sides are the sine and cosine values themselves. Thus, the squares of these sides sum to 1 just as in the Pythagorean theorem:\[sin^2 x + cos^2 x = 1\]In the exercise, this identity was crucial in transforming the expression \(1 - \sin^2 x\) into \(\cos^2 x\), aiding in the simplification of the equation.

Sine and Cosine

Sine and cosine are two of the primary trigonometric functions. They originate from the unit circle, and understanding them is essential for mastering trigonometry.

• Sine: The sine of an angle is the y-coordinate of the corresponding point on the unit circle. In a right triangle, it is the ratio of the opposite side to the hypotenuse.
• Cosine: The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. In a right triangle, it is the ratio of the adjacent side to the hypotenuse.

Using these definitions, each trigonometric identity can be rewritten in terms of sine and cosine. This is particularly useful because it allows you to express complex trigonometric functions more simply. In the exercise, expressing \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\) in terms of sine and cosine paved the way for further simplifications.

Cosecant and Cotangent

Cosecant and cotangent are the reciprocal identities of sine and tangent. Understanding these will help to solve many trigonometric problems more efficiently.

• Cosecant (\(\csc\)): Defined as the reciprocal of sine: \(\csc x = \frac{1}{\sin x}\). It is undefined wherever sine is zero, this depends on values of \(x\) such that \(\sin x = 0\).
• Cotangent (\(\cot\)): Related to tangent, cotangent is its reciprocal. Thus, \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\). It is undefined wherever sine is zero as well.

These reciprocal identities simplify problems and expressions, as shown in the step-by-step solution where \(\csc x\) and \(\cot x\) were rewritten in terms of the more basic functions sine and cosine. This allowed the exercise to reduce complex expressions into more manageable forms.

Equation Simplification

Equation simplification is an essential skill in solving trigonometric identities and equations. It involves rewriting expressions to make them easier to work with.
In trigonometry, this often means using identities and algebraic manipulation. The step-by-step solution of the exercise is a perfect example of effective equation simplification:

• Combine Fractions: Initially, the left side of the equation was expressed as \(\frac{1}{\sin x} - \sin x\). Combining these into a single fraction, \(\frac{1 - \sin^2 x}{\sin x}\), was a crucial step.
• Use Identities: Applying the Pythagorean identity to replace \(1 - \sin^2 x\) with \(\cos^2 x\) simplified the expression considerably.
• Factorization: By recognizing that \(\cos x \cdot \frac{\cos x}{\sin x}\) equals \(\cos x \cot x\), the equation was verified to be true.

Each of these steps illustrates how breaking a problem into smaller, more manageable pieces can lead to a clear and correct solution despite initially appearing complex.