Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \sec \theta \csc \theta=\tan \theta+\cot \theta $$

Step-by-Step Solution

Answer

Both sides of the equation simplify to \( \frac{1}{\cos \theta \sin \theta} \), proving the identity.

Step 1: Convert to Sine and Cosine

First, express all trigonometric functions in terms of sine and cosine. We know that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, the left side of the equation becomes \( \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\cos \theta \sin \theta} \).

Step 2: Expand Right Side

Now, let's express \( \tan \theta \) and \( \cot \theta \) in terms of sine and cosine. We have \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, the right side of the equation becomes \( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \).

Step 3: Find a Common Denominator

To add the fractions on the right side, \( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \), we need a common denominator. The common denominator for these fractions is \( \cos \theta \sin \theta \).

Step 4: Combine Fractions

Rewrite each term with the common denominator: \( \frac{\sin^2 \theta}{\cos \theta \sin \theta} + \frac{\cos^2 \theta}{\sin \theta \cos \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta} \).

Step 5: Apply Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute this into the equation: \( \frac{1}{\cos \theta \sin \theta} \).

Step 6: Compare Both Sides

Now both sides of the equation are \( \frac{1}{\cos \theta \sin \theta} \). Hence, the identity is verified.

Key Concepts

Secant

In trigonometry, the secant function is one of the six fundamental trigonometric functions. It is commonly abbreviated as "sec". The secant of an angle in a right triangle is defined as the reciprocal of the cosine of that angle. This means that

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding "sec" is crucial for solving trigonometric identities and equations as it often appears in transformations and simplifications.
When working on identities, expressing secant in terms of sine and cosine can simplify things. Like in this exercise, expressing secant as \( \frac{1}{\cos \theta} \) helps us compare both sides of the identity easily. Remember, the key to mastering trigonometric functions is practice and consistent use in various problems.

Cosecant

The cosecant, abbreviated as "csc", is another reciprocal trigonometric function, just like secant. Cosecant is the reciprocal of sine. In mathematical terms, this is expressed as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

In identities, the use of cosecant helps by simplifying or breaking down expressions.
This function, as demonstrated above, plays a vital role alongside secant when proving certain identities. For instance, converting the equations into terms of sine and cosine, where cosecant becomes \( \frac{1}{\sin \theta} \), can often make complex identities more manageable. Practice understanding how to switch between different trigonometric forms with ease.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstones of trigonometry. This identity connects sine and cosine in a fundamental way and is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

The beauty of this relationship is its universality in trigonometry. It's utilized to simplify expressions and prove equations, as seen in our exercise.
In the solution, replacing \( \sin^2 \theta + \cos^2 \theta \) with 1 on the right side of the identity made it possible to equate it with the left side. Recognizing and applying this identity in various contexts is an essential skill. Remember, this identity will often help in finding missing trigonometric values and verifying identities.

Sine and Cosine Conversion

Converting expressions into sine and cosine is a common strategy to simplify and solve trigonometric equations. The ability to switch between different trigonometric forms can make many problems significantly more approachable.

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In our exercise, both sides were converted to sine and cosine, revealing their simple similarities.
This technique is especially useful because it reduces the number of symbols in your work and often leads to simpler overall expression or equation manipulation. It shows how different trigonometric functions relate to one another and streamlines solving identities or equations in trigonometry.