Problem 35 - Algebra and Trigonometry

Question

Verify the identity. $$ \tan \theta+\cot \theta=\sec \theta \csc \theta $$

Step-by-Step Solution

Answer

The identity \( \tan \theta + \cot \theta = \sec \theta \csc \theta \) is verified.

Step 1: Rewrite Tangent and Cotangent

Write the tangent (\(\tan \theta\) ) and cotangent (\(\cot \theta\)) in terms of sine and cosine: \[\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}.\]

Step 2: Combine Left Side Under a Common Denominator

Combine \( \tan \theta + \cot \theta \) under a common denominator. The common denominator of \( \frac{\sin \theta}{\cos \theta} \) and \( \frac{\cos \theta}{\sin \theta} \) is \( \sin \theta \cos \theta \). So: \[ \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}. \]

Step 3: Simplify Using Pythagorean Identity

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to simplify the expression: \[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}. \]

Step 4: Rewrite the Right Side

Rewrite \(\sec \theta \csc \theta\) using their definitions: \[ \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}. \] Therefore, \[ \sec \theta \csc \theta = \frac{1}{\cos \theta} \times \frac{1}{\sin \theta} = \frac{1}{\sin \theta \cos \theta}. \]

Step 5: Verify the Identity

Compare the expressions from Steps 3 and 4: both sides are equal to \( \frac{1}{\sin \theta \cos \theta} \). Thus, the identity is verified: \( \tan \theta + \cot \theta = \sec \theta \csc \theta \).

Key Concepts

Tangent and Cotangent

The tangent and cotangent functions are fundamental in trigonometry. - **Tangent** is defined as the ratio of the sine function to the cosine function. Mathematically, it is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This tells us that tangent measures the slope of the angle \( \theta \) on the unit circle.- **Cotangent** is the reciprocal of the tangent function. Its formula is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Essentially, it provides the reciprocal slope in relation to the tangent.Understanding these functions in terms of sine and cosine helps simplify many trigonometric identities and equations. Converting tangent and cotangent into these ratios was crucial in verifying the identity \( \tan \theta + \cot \theta = \sec \theta \csc \theta \) by expressing them in terms of sine and cosine to facilitate manipulation and combination.

Sine and Cosine

The sine and cosine functions are the backbone of trigonometry. - The **sine function** is associated with the y-coordinate on the unit circle and is represented simply as \( \sin \theta \). It indicates the vertical position of a point on the circle.- The **cosine function** relates to the x-coordinate on the unit circle, given as \( \cos \theta \). This shows the horizontal position of a point on the circle.These functions allow the expression of other trigonometric functions, such as tangent and cotangent.

• For example, using them helps rewrite tangent as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), capturing its essence as a ratio of these two primary axes' distances.
• Similarly, cotangent is expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Converting each function to a ratio of sine and cosine facilitates algebraic manipulation and verifies identities or simplifies expressions.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry, connecting sine and cosine functions.- It states that the square of the sine of an angle plus the square of the cosine of the same angle equals one: \( \sin^2 \theta + \cos^2 \theta = 1 \).This identity mirrors the Pythagorean theorem, hence its name. In verifying the trigonometric identity \( \tan \theta + \cot \theta = \sec \theta \csc \theta \), it plays a pivotal role:

• In Step 3 of the solution, the identity simplifies the numerator \( \sin^2 \theta + \cos^2 \theta \) to 1 when combining the tangent and cotangent expressions under a common denominator.

Using the Pythagorean identity effectively reduces complex expressions and is an essential tool in proving various trigonometric equations.