Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sec 35^{\circ} \csc 55^{\circ}-\tan 35^{\circ} \cot 55^{\circ}$$

Short answer

1

Step-by-step solution

Recognize Complementary Angles

Notice that 35° and 55° are complementary angles since their sum is 90°. Use this fact to rewrite the trigonometric functions.

Apply Complementary Angle Theorem

Use the Complementary Angle Theorem which states that for any angle θ, \(\text{sec}(90^{\text{°}} - θ) = \text{csc}(θ)\) and \(\text{tan}(90^{\text{°}} - θ) = \text{cot}(θ)\). Rewrite the expressions: \(\text{sec}(35^{\text{°}}) = \text{csc}(55^{\text{°}})\) and \(\tan(35^{\text{°}}) = \text{cot}(55^{\text{°}})\).

Substitute and Simplify

Substitute the complementary angles into the original expression: \(\text{sec}(35^{\text{°}}) \text{csc}(55^{\text{°}}) - \text{tan}(35^{\text{°}}) \text{cot}(55^{\text{°}})\). This becomes \( \text{csc}(55^{\text{°}}) \text{csc}(55^{\text{°}}) - \text{cot}(55^{\text{°}}) \text{cot}(55^{\text{°}}) \).

Apply Trigonometric Identities

Recall the trigonometric identities: \( \text{csc}(θ) = \frac{1}{\text{sin}(θ)} \) and \( \text{cot}(θ) = \frac{1}{\text{tan}(θ)} \). Simplify the expression to: \( \frac{1}{\text{sin}(55^{\text{°}})} \frac{1}{\text{sin}(55^{\text{°}})} - \frac{1}{\text{tan}(55^{\text{°}})} \frac{1}{\text{tan}(55^{\text{°}})} \).

Simplify Further

Combine and simplify the products: \( \frac{1}{\text{sin}^2(55^{\text{°}})} - \frac{1}{\text{tan}^2(55^{\text{°}})} \). Use the identity \( \text{tan}(θ) = \frac{\text{sin}(θ)}{\text{cos}(θ)} \) to further simplify: \( \frac{1}{\text{sin}^2(55^{\text{°}})} - \frac{\text{cos}^2(55^{\text{°}})}{\text{sin}^2(55^{\text{°}})} \).

Final Simplification

Combine the fractions: \( \frac{1 - \text{cos}^2(55^{\text{°}})}{\text{sin}^2(55^{\text{°}})} \). Use the Pythagorean identity \( 1 - \text{cos}^2(θ) = \text{sin}^2(θ) \) to simplify: \( \frac{\text{sin}^2(55^{\text{°}})}{\text{sin}^2(55^{\text{°}})} = 1 \).

Key concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. For example, 35° and 55° are complementary because 35° + 55° = 90°. Understanding complementary angles is essential in trigonometry because certain trigonometric functions of complementary angles are equal. For instance, if two angles θ and 90° - θ are complementary, then \(\text{sin}(θ) = \text{cos}(90° - θ)\), and \(\text{tan}(θ) = \text{cot}(90° - θ)\).

This relationship helps to transform trigonometric expressions and can simplify complex problems without a calculator. When you become familiar with these transformations, it makes solving trigonometric problems much easier.

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. The most commonly used fundamental identities include:

• Sine and cosine: \(\text{sin}(θ)\) and \(\text{cos}(θ)\)
• Tangent and cotangent: \(\text{tan}(θ) = \frac{\text{sin}(θ)}{\text{cos}(θ)}\) and \(\text{cot}(θ) = \frac{\text{cos}(θ)}{\text{sin}(θ)}\)
• Secant and cosecant: \(\text{sec}(θ) = \frac{1}{\text{cos}(θ)}\) and \(\text{csc}(θ) = \frac{1}{\text{sin}(θ)}\)

These identities are crucial because they form the basis for manipulating and simplifying trigonometric expressions. In the example provided, recognizing that \(\text{sec}(35^{°})\) can be rewritten as \(\text{csc}(55^{°})\) using the identity property of complementary angles directly simplifies the expression.

Pythagorean Identities

The Pythagorean identities are derived from the Pythagorean theorem and are essential in trigonometry. They are:

• \(\text{sin}^2(θ) + \text{cos}^2(θ) = 1\)
• \(1 + \text{tan}^2(θ) = \text{sec}^2(θ)\)
• \(1 + \text{cot}^2(θ) = \text{csc}^2(θ)\)

In the provided solution, we used the first Pythagorean identity: \(\text{sin}^2(55^{°}) + \text{cos}^2(55^{°}) = 1\).

This identity simplifies the expression \( \frac{1 - \text{cos}^2(55^{°})}{\text{sin}^2(55^{°})}\) to \(\frac{\text{sin}^2(55^{°})}{\text{sin}^2(55^{°})} = 1\).
This application shows how powerful and necessary these identities are for solving trigonometric problems efficiently.