Problem 22 - Contemporary Precalculus

Question

Prove the identity. $$\tan x(\cos x+\csc x)=\sin x+\sec x$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity $$\tan x(\cos x+\csc x)=\sin x+\sec x$$ Solution: 1. Write the secant and cosecant functions in terms of sine and cosine: $$\tan x \left(\cos x + \frac{1}{\sin x}\right) = \sin x + \frac{1}{\cos x}$$ 2. Rewrite the tangent function in terms of sine and cosine: $$\frac{\sin x}{\cos x} \left(\cos x + \frac{1}{\sin x}\right) = \sin x + \frac{1}{\cos x}$$ 3. Distribute the term on the left-hand side: $$\frac{\sin x}{\cos x} \cdot \cos x + \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} = \sin x + \frac{1}{\cos x}$$ 4. Simplify terms: $$\sin x + \frac{1}{\cos x} = \sin x + \frac{1}{\cos x}$$ 5. Conclusion: We have proven the trigonometric identity $$\tan x(\cos x+\csc x)=\sin x+\sec x$$

Step 1: Rewrite trigonometric functions in terms of sine and cosine

Start by rewriting the secant and cosecant functions in terms of sine and cosine. Recall that \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\). Rewrite the given identity as: $$\tan x \left(\cos x + \frac{1}{\sin x}\right) = \sin x + \frac{1}{\cos x}$$

Step 2: Rewrite tangent function in terms of sine and cosine

Recall that \(\tan x = \frac{\sin x}{\cos x}\). Substitute this into the left-hand side of the identity: $$\frac{\sin x}{\cos x} \left(\cos x + \frac{1}{\sin x}\right) = \sin x + \frac{1}{\cos x}$$

Step 3: Distribute the term on the left-hand side

Next, distribute the \(\frac{\sin x}{\cos x}\) term on the left-hand side: $$\frac{\sin x}{\cos x} \cdot \cos x + \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} = \sin x + \frac{1}{\cos x}$$

Step 4: Simplify terms

Simplify each term on the left-hand side: $$\sin x + \frac{1}{\cos x} = \sin x + \frac{1}{\cos x}$$

Step 5: State the conclusion

We have shown that the left-hand side of the given identity can be simplified to the right-hand side, thus proving the identity: $$\tan x(\cos x+\csc x)=\sin x+\sec x$$

Key Concepts

Sine and Cosine Functions

Understanding sine and cosine functions is crucial in trigonometry. They are the building blocks of many other trigonometric functions. The sine function, denoted as \( \sin x \), calculates the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function, denoted as \( \cos x \), calculates the ratio of the adjacent side to the hypotenuse.

Both sine and cosine are periodic functions, meaning they repeat their values at regular intervals. They share fundamental properties such as range and period:

• Range: For both \( \sin x \) and \( \cos x \), the output is always between \(-1\) and \(1\).
• Period: Both functions have a period of \(2\pi\), meaning \( \sin(x + 2\pi) = \sin x \) and \( \cos(x + 2\pi) = \cos x \).

A key identity combining these functions is \( \sin^2 x + \cos^2 x = 1 \), a cornerstone in proving other trigonometric identities.

Tangent Function

The tangent function, usually represented as \( \tan x \), is another key function in trigonometry. It is defined by the ratio of the sine function to the cosine function, i.e., \( \tan x = \frac{\sin x}{\cos x} \). This fundamental relationship links the tangent directly to sine and cosine, providing flexibility in rewriting expressions and solving equations.

The tangent function has some unique characteristics:

• It is undefined when \( \cos x = 0 \) because division by zero occurs.
• The range of \( \tan x \) is all real numbers, and it has a period of \(\pi\), meaning \( \tan(x + \pi) = \tan x \).

The function is helpful in solving geometry problems where you need to connect angles and side lengths, especially in non-right triangles.

Secant and Cosecant Functions

Secant and cosecant are the reciprocal functions of cosine and sine, respectively. Understanding their relationships is essential for proving complex identities.

Secant, denoted as \( \sec x \), is the reciprocal of cosine:

• \( \sec x = \frac{1}{\cos x} \)

Cosecant, denoted as \( \csc x \), is the reciprocal of sine:

• \( \csc x = \frac{1}{\sin x} \)

Both functions are undefined where their respective base functions are zero due to division by zero.

These reciprocal relations are critical when rewriting expressions for trigonometric proofs, helping simplify equations or combine terms.

Trigonometric Proofs

Trigonometric proofs require a thorough understanding of identities and relationships between trigonometric functions. They're like solving puzzles by manipulating components to verify an equality. The identity \( \tan x(\cos x+\csc x)=\sin x+\sec x \) is one such example that involves transforming and simplifying expressions.

To prove an identity:

• Start by expressing all trigonometric functions in terms of sine and cosine, which provides uniformity.
• Use fundamental identities like \( \tan x = \frac{\sin x}{\cos x} \), \( \sec x = \frac{1}{\cos x} \), and \( \csc x = \frac{1}{\sin x} \) to rewrite the expressions.
• Simplify both sides step-by-step to reach the same expression or value.

Practice is key to mastering trigonometric proofs, as it develops familiarity with common manipulation techniques and identities.