In Exercises 23–26, find the exact value of each expression. Do not use a calculator.cos² 𝜋 - tan² 𝜋 4 4

Verified step by step guidance

1. Identify the trigonometric values: \( \cos\left(\frac{\pi}{4}\right) \) and \( \tan\left(\frac{\pi}{4}\right) \).
2. Recall that \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \tan\left(\frac{\pi}{4}\right) = 1 \).
3. Square the values: \( \cos^2\left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^2 \) and \( \tan^2\left(\frac{\pi}{4}\right) = 1^2 \).
4. Substitute the squared values into the expression: \( \cos^2\left(\frac{\pi}{4}\right) - \tan^2\left(\frac{\pi}{4}\right) \).
5. Simplify the expression to find the exact value.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as cosine (cos) and tangent (tan), relate angles to ratios of sides in right triangles. The cosine function gives the ratio of the adjacent side to the hypotenuse, while the tangent function is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for evaluating expressions involving angles.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin²(θ) + cos²(θ) = 1 holds true. This identity is fundamental in trigonometry and can be manipulated to express one function in terms of another. It is particularly useful when simplifying expressions involving squares of trigonometric functions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions for specific angles (like 0, π/6, π/4, π/3, and π/2) are often derived from the unit circle. For example, cos(π) = -1 and tan(π) = 0. Knowing these exact values allows for precise calculations without the need for a calculator, which is crucial for solving problems that require exact answers.