Problem 46 - Algebra 2

Question

Verify each identity. $$ \sec \theta-\sin \theta \tan \theta=\cos \theta $$

Step-by-Step Solution

Answer

The given trigonometric identity \(\sec \theta-\sin \theta \tan \theta=\cos \theta\) has been verified.

Step 1: Convert all functions to sine and cosine

The secant function, \(\sec \theta\), can be written as \(1/\cos\theta\), and the tangent function, \(\tan\theta\), can be written as \(\sin\theta/\cos\theta\). Let's substitute these in to simplify the left side of the equation: \((1/\cos\theta)-\sin\theta*(\sin\theta/\cos\theta)\)

Step 2: Simplify the equation

Simplify the equation by performing the multiplication: \((1/\cos\theta)-(\sin^2\theta/\cos\theta)\). Now since the denominators are the same, we can combine the fractions: \((1-\sin^2\theta)/\cos\theta\)

Step 3: Use the Pythagorean Identity

By Pythagorean Identity, we know that \(1-\sin^2\theta = \cos^2\theta\). So, substituting \(\cos^2\theta\) for \(1-\sin^2\theta\), we get: \(\cos^2\theta/\cos\theta\)

Step 4: Simplify the expression

Finally, simplify \(\cos^2\theta/\cos\theta\) to \(\cos\theta\). Now the right side of the equation matches the left side, verifying the identity.

Key Concepts

Secant Function

The secant function is a fundamental trigonometric function that is the reciprocal of the cosine function. In other words, \[\sec \theta = \frac{1}{\cos \theta}.\]Understanding the secant function helps in simplifying trigonometric expressions, just like in the exercise.

• When dealing with complex trigonometric identities, converting the secant to cosine can simplify the problem. This is because cosine is a more foundational trigonometric function compared to secant.
• When you rewrite the secant function in terms of cosine, as in \(\sec \theta = \frac{1}{\cos \theta}\), it becomes easier to combine and simplify expressions with common denominators.

Learning to see the relationships between these functions can enhance your ability to manipulate and solve trigonometric equations. The secant function can appear daunting, but its simplification to cosine makes it much more approachable.

Tangent Function

The tangent function is another core trigonometric function. It represents the ratio of the sine to the cosine of the same angle. This means:\[\tan \theta = \frac{\sin \theta}{\cos \theta}.\]This relationship makes tangent an essential tool for understanding trigonometric identities.

• Tangent can be quite helpful in simplifying expressions because it combines both sine and cosine into a single function.
• As seen in the exercise, rewriting the tangent function helps in breaking down and simplifying complex identities. By converting it into \(\frac{\sin \theta}{\cos \theta}\), it aligns the expression into a common denominator, allowing for straightforward simplifications.

Recognizing when to rewrite tangent in terms of sine and cosine is a strategic approach that simplifies problem-solving in trigonometry.

Pythagorean Identity

The Pythagorean Identity is one of the most vital identities in trigonometry. It relates the square of sine and cosine to the value 1:\[\sin^2 \theta + \cos^2 \theta = 1.\]This identity underpins many aspects of trigonometric simplifications, such as those seen in verifying identities.

• As demonstrated in the exercise, this identity allows you to replace expressions like \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), streamlining the process of simplifying fractions and expressions.
• It's often employed to convert between sine and cosine in proofs and simplifications, making life simpler when solving complex trig equations.

Mastering the use of the Pythagorean Identity is crucial. It allows conversions that lead to simpler forms and solutions in trigonometric identities, providing shortcuts and direct pathways to solutions.

Sine and Cosine Functions

Sine and cosine are the two foundational trigonometric functions traceable to the unit circle. Each function's value relates directly to a point on this circle, with:

• The sine function representing the y-coordinate and
• The cosine function representing the x-coordinate of a point as an angle \(\theta\) changes.

Understanding these functions is essential because:

• They form the basis from which other trigonometric functions, like tangent and secant, are derived.
• They are the building blocks for converting more complex functions into simpler identities to prove or solve trigonometric equations.

In the context of identities, such as the one in the exercise where everything is eventually expressed in terms of \(\cos \theta\), the sine and cosine functions provide a framework to break down and simplify the equations. A firm grasp of how sine and cosine work leads to an easier understanding of how to manipulate trigonometric expressions for solutions.