Problem 6 - Algebra 2

Question

Verify each identity. $$ \sec \left(\frac{\pi}{2}-\theta\right)=\csc \theta $$

Step-by-Step Solution

Answer

The identity \(\sec \left(\frac{\pi}{2}-\theta\right)=\csc \theta\) is verified by converting it into an equation where the left-hand side and the right-hand side are both equal to \(\frac{1}{\sin \theta}\).

Step 1: Rewrite in terms of sine and cosine

Rewrite the given equation in terms of sine and cosine using reciprocal identities: \(\frac{1}{\cos \left(\frac{\pi}{2}-\theta\right)}=\frac{1}{\sin \theta}\)

Step 2: Apply co-function identities

Replace the cosine function in the left side of the equation with the sine function using the co-function identity (since \(\cos(\frac{\pi}{2}-\theta)=\sin \theta\)): \(\frac{1}{\sin \theta}=\frac{1}{\sin \theta}\)

Step 3: Verify the identity

As both sides of the equation are now identical, the identity has been verified.

Key Concepts

Co-Function Identities

In trigonometry, understanding co-function identities is vital for simplifying expressions and verifying identities. Co-function identities relate functions of complementary angles. Two angles are complementary if their sum is \( \frac{\pi}{2} \) radians (or 90 degrees). For instance, the co-function identity for cosine states that \( \cos \left( \frac{\pi}{2} - \theta \right) = \sin \theta \).
Thus, when you see \( \cos \left( \frac{\pi}{2} - \theta \right) \), you can directly replace it with \( \sin \theta \). This transformation is very handy and is frequently used in problems involving trigonometric identities.

• The paired functions are sine & cosine, tangent & cotangent, and secant & cosecant.
• The co-function identities allow you to interchange functions for complementary angles.

By using these identities, complex trigonometric expressions can be broken down into simpler forms, making it easier to understand and verify them.

Reciprocal Identities

Reciprocal identities express trigonometric functions in terms of their reciprocals. Understanding these identities is essential for transforming and simplifying trigonometric expressions. Here are the fundamental reciprocal identities:

• The cosine of an angle \( \theta \) is the reciprocal of the secant: \( \cos \theta = \frac{1}{\sec \theta} \).
• Similarly, sine and cosecant are reciprocals: \( \sin \theta = \frac{1}{\csc \theta} \).
• Tangent and cotangent are also reciprocals: \( \tan \theta = \frac{1}{\cot \theta} \).

These identities are crucial when expressions involve multiplicative inverse functions. In the provided exercise, using reciprocal identities helped rewrite \( \sec \) and \( \csc \) in terms of cosine and sine, setting the stage for further simplification. This ability to transform functions is vital for solving and verifying identities efficiently.

Verifying Identities

The process of verifying identities in trigonometry involves proving that two sides of an equation are equal by using known identities and algebraic manipulation. This ensures the equation holds true for all permitted values of the variables involved.
The exercise demonstrates this by equating \( \sec \left( \frac{\pi}{2} - \theta \right) \) to \( \csc \theta \). The verification process involved the following steps:

• Rewrite each function using identities (such as reciprocal and co-function identities).
• Simplify both sides of the equation as needed.
• Gently transform expressions into identical forms.

By showing that the left-hand side equaled the right-hand side after simplification, the identity was verified. Verifying identities strengthens your understanding of trigonometric properties and improves problem-solving skills, aiding in tackling more complex mathematical challenges.