91Ó°ÊÓ

In trigonometry, fundamental identities are equations that hold true for all values of the included variables. They are essential for solving problems and simplifying expressions. One of the most important fundamental identities is the Pythagorean identity: \( \text{sin}^2(\theta) + \text{cos}^2(\theta) = 1 \). This identity helps us in verifying and deriving other identities.
Another key identity is the ratio identity for tangent, which is given by \( \tan(\theta) = \frac{ \text{sin}(\theta) }{ \text{cos}(\theta) } \). This is particularly helpful when we need to express tangent in terms of sine and cosine.
Additionally, there are the reciprocal identities:

• \( \text{csc}(\theta) = \frac{1}{ \text{sin}(\theta) } \)
• \( \text{sec}(\theta) = \frac{1}{ \text{cos}(\theta) } \)
• \( \text{cot}(\theta) = \frac{1}{ \text{tan}(\theta) } \)

These identities provide the basis for transforming and simplifying trigonometric expressions effectively.

Trigonometric functions are functions of an angle and are essential to various fields of mathematics and science. The basic trigonometric functions include sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}).

• The sine function relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse: \( \text{sin}(\theta) = \frac{ \text{opposite} }{ \text{hypotenuse} } \).
• The cosine function relates the angle to the ratio of the length of the adjacent side to the hypotenuse: \( \text{cos}(\theta) = \frac{ \text{adjacent} }{ \text{hypotenuse} } \).
• The tangent function is the ratio of the opposite to adjacent side: \( \text{tan}(\theta) = \frac{ \text{opposite} }{ \text{adjacent} } \).

These functions help in understanding and solving problems related to angles and lengths in various geometric settings.
In our problem, knowing that \( \tan(70^\bullet) = \frac{1}{ \tan(20^\bullet) } \) allows us to use the Complementary Angle Theorem to simplify the expression.

Exact values of trigonometric functions are specific values at particular angles that are known and can be used directly without a calculator. For example, the exact values for common angles like 0°, 30°, 45°, 60°, and 90° are well-documented and memorized.
For instance:

• \( \text{sin}(30^\bullet) = \frac{1}{2} \)
• \( \text{cos}(30^\bullet) = \frac{\sqrt{3}}{2} \)
• \( \text{tan}(45^\bullet) = 1 \)

These values are derived from the properties of equilateral and isosceles right triangles.
In the given problem, recognizing exact values and identities, coupled with the complementary angles, simplifies finding the value of \( \tan(20^\bullet) - \frac{ \text{cos}(70^\bullet) }{ \text{cos}(20^\bullet) } \). Using the fact that \( \text{cos}(70^\bullet) = \text{sin}(20^\bullet) \), and \( \text{tan}(70^\bullet) = \frac{1}{ \text{tan}(20^\bullet) } \), aids in transforming the expression to equivalent forms with known values.