Problem 12 - Algebra and Trigonometry

Question

Verify each identity. \(\tan \theta+\cot \theta=\sec \theta \csc \theta\)

Step-by-Step Solution

Answer

The identity \(\tan \theta+\cot \theta=\sec \theta \csc \theta \) holds true for all values of \(\theta\) as long as \(\cos \theta\) and \(\sin \theta\) are non zero.

Step 1: Express everything in terms of sine and cosine

Use the definitions of tangent, cotangent, secant, and cosecant to replace each of the terms in the expression: \(\tan \theta +\cot \theta = \sec \theta \csc \theta \) turns into \(\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} \)

Step 2: Simplify the right side

Simplify the right side of the equation. Since both terms in the right side of the equation are in fractions and their denominators are being multiplied, you can apply the rule of multiplication of fractions: \(\frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\cos \theta \sin \theta}\)

Step 3: Find common denominator for the left side

To add fractions, they must have a common denominator. Apply the fundamental rule \(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}\). The left side turns into \(\frac{\sin^2 \theta +\cos^2 \theta}{\cos \theta \sin \theta}\)

Step 4: Recognize and use the Pythagorean identity

Recall the identity \(\sin^2 \theta +\cos^2 \theta = 1\). Replacing the numerator with \(1\), the left side becomes \(\frac{1}{\cos \theta \sin \theta}\)

Step 5: Compare the results

The left side \(\frac{1}{\cos \theta \sin \theta}\) is now equal to the right side \(\frac{1}{\cos \theta \sin \theta}\), which completes the verification.

Key Concepts

Tangent and Cotangent

Understanding the relationship between tangent and cotangent is crucial for simplifying complex trigonometric expressions. Tangent, denoted as \( \tan \theta \), is a ratio of the opposite side to the adjacent side in a right-angled triangle, or in terms of sine and cosine, it is \(\frac{\sin \theta}{\cos \theta}\). Cotangent, denoted as \( \cot \theta \), is the reciprocal of tangent, which can also be expressed as the ratio of the adjacent side to the opposite side, or \(\frac{\cos \theta}{\sin \theta}\) in terms of sine and cosine.

When simplifying expressions involving \(\tan\) and \(\cot\), remember that they can be converted into \(\sin\) and \(\cos\) to find a common denominator or simplify fractions. This method was applied in the exercise to combine these trigonometric functions into one expression.

Secant and Cosecant

Secant (\(\sec \theta\)) and cosecant (\(\csc \theta\)) are less commonly used trigonometric functions but are just as important. In terms of a right-angled triangle, the secant represents the ratio of the hypotenuse to the adjacent side, and cosecant is the ratio of the hypotenuse to the opposite side. Algebraically, \(\sec \theta\) is the reciprocal of \(\cos \theta\) and \(\csc \theta\) is the reciprocal of \(\sin \theta\).

The product of secant and cosecant gives us a new expression \(\sec \theta \times \csc \theta\) which is equivalent to \(\frac{1}{\cos \theta \sin \theta}\). This transformed expression was used in the step by step solution to show the equivalence on both sides of the identity.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect of trigonometry and is derived from the Pythagorean theorem. It states that for any angle \(\theta\), the square of the sine plus the square of the cosine equals one: \(\sin^2 \theta + \cos^2 \theta = 1\).

This identity is especially useful when simplifying trigonometric expressions involving \(\sin^2 \theta\) or \(\cos^2 \theta\). For instance, in the provided exercise, the numerator of the left side of the equation transformed into 1 thanks to this identity. Recognizing such relationships allows simplification of complex trigonometric expressions and proves various trigonometric identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves several techniques including finding common denominators, converting functions into sine and cosine, and applying fundamental identities like the Pythagorean identity. Breaking down complex expressions into simpler terms by finding common denominators, as was shown in the exercise, can greatly aid in combining fractions.

Also crucial is the understanding and application of trigonometric identities. These identities are tools that allow you to transform and simplify expressions which may appear intimidating at first. In the exercise, you can see how converting tangent and cotangent into their sine and cosine forms and using the Pythagorean identity simplifies the verification process of the given identity.