Problem 62 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x \).

Step 1: Convert to Basic Trigonometric Functions

The problem asks us to verify an identity that involves secant, cosecant, tangent, and cotangent. We start by expressing these in terms of sine and cosine. Recall that \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), \( \tan x = \frac{\sin x}{\cos x} \), and \( \cot x = \frac{\cos x}{\sin x} \). Substitute these into the equation:\[\frac{\frac{1}{\cos x} + \frac{1}{\sin x}}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}}\]

Step 2: Simplify the Numerator

The numerator is \( \frac{1}{\cos x} + \frac{1}{\sin x} \). Combine these into a single fraction:\[\frac{\sin x + \cos x}{\sin x \cos x}\]

Step 3: Simplify the Denominator

The denominator is \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \). Combine these into a single fraction by finding a common denominator:\[\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\]Note that \( \sin^2 x + \cos^2 x = 1 \), the Pythagorean identity. Thus, the denominator simplifies to:\[\frac{1}{\sin x \cos x}\]

Step 4: Divide Numerator by Denominator

We have:\[\frac{\frac{\sin x + \cos x}{\sin x \cos x}}{\frac{1}{\sin x \cos x}}\]To perform the division, multiply by the reciprocal of the denominator:\[(\sin x + \cos x) \times 1 = \sin x + \cos x\]

Step 5: Conclude the Identity Verification

The expression simplifies to \( \sin x + \cos x \), which is exactly the right side of the given identity. Thus, we have verified that:\[\frac{\sec x + \csc x}{\tan x + \cot x} = \sin x + \cos x\]

Key Concepts

Secant and Cosecant

The secant and cosecant functions are reciprocals of the basic trigonometric functions cosine and sine, respectively. This means they offer alternative ways to express trigonometric ratios that might be more convenient in certain calculations. The secant, denoted as \(\sec x\), is defined as \(\frac{1}{\cos x}\), while the cosecant, denoted as \(\csc x\), is defined as \(\frac{1}{\sin x}\).
These ratios are particularly useful when dealing with identities and transformations in trigonometry. They help in converting complex trigonometric expressions into a form that is easier to simplify or compare.

• Secant - Reciprocates the cosine value. If \(\cos x = 0\), then \(\sec x\) is undefined because dividing by zero is not allowed.
• Cosecant - Reciprocates the sine value. Similar to secant, if \(\sin x = 0\), \(\csc x\) is undefined.

Tangent and Cotangent

Tangent and cotangent offer different perspectives on the relationships between the sides of a right triangle. The tangent of an angle, represented as \(\tan x\), is defined as the ratio of the opposite side to the adjacent side, which can also be written in terms of sine and cosine as \(\tan x = \frac{\sin x}{\cos x}\).
The cotangent, on the other hand, is the reciprocal of the tangent, given by \(\cot x = \frac{\cos x}{\sin x}\).
These functions are often used to simplify trigonometric equations, and their interrelationship with sin and cos can be pivotal in proving identities.

• Tangent - Represents the slope of the angle in the curve of the circle when expressed as a sine over cosine ratio.
• Cotangent - As reciprocal to tangent, providing another angle's slope perspective.

Sine and Cosine

The sine and cosine functions are the fundamental building blocks of trigonometry. They define the relationships between angles and the lengths of sides in right triangles. Specifically, the sine of an angle \(\sin x\) is the length of the opposite side over the hypotenuse, and the cosine \(\cos x\) is the length of the adjacent side over the hypotenuse.
Performing operations and simplifications in trigonometry often revolves around expressing complex functions in terms of sine and cosine. Their periodic nature and ability to transform through identities make them incredibly versatile.

• Sine - Measures the vertical component in circular motion or angle.
• Cosine - Measures the horizontal component in circular motion or angle.

Pythagorean Identity

In trigonometry, one of the most crucial relationships is the Pythagorean identity, which states that \(\sin^2 x + \cos^2 x = 1\). This identity stems from the Pythagorean theorem and describes the inherent relationship between the sine and cosine functions.
This identity is particularly powerful because it allows for the conversion and simplification of expressions involving sine and cosine into alternative forms. When verifying identities, such as in the original exercise, recognizing and applying the Pythagorean identity can often lead to substantial simplifications that expose the underlying structure of the expression.

• Serves as the bedrock for many trigonometric identities.
• Useful in converting complex terms back to basic trigonometric forms.