Problem 38 - Algebra 2

Question

Verify each identity. $$ \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} $$

Step-by-Step Solution

Answer

Following the above steps which involve applying the identities for the difference of angles in sine and cosine, recalling the definition of tangent and then simplifying appropriately gives us \( \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B} \), hence the original trigonometric identity is verified. It's of great importance to remember the fundamental trigonometric identities as they form the basis for proving more complex equations.

Step 1: Difference of Angles in Sine and Cosine

Recall the identities for sine and cosine when you have the difference of angles. \( \sin(A-B)=\sin A\cos B-\cos A\sin B \) and \( \cos(A-B)=\cos A\cos B+\sin A\sin B \)

Step 2: Definition of Tangent

Recall also the definition of tangent: \( \tan x=\frac{\sin x}{\cos x} \). But for \( A-B \), this becomes \( \tan(A-B)=\frac{\sin(A-B)}{\cos(A-B)} \). Substitute the identities from step 1, resulting in \( \tan(A-B)=\frac{\sin A\cos B-\cos A\sin B}{\cos A\cos B+\sin A\sin B} \).

Step 3: Simplification

Divide both the numerator and the denominator of the right-hand side by \( \cos A\cos B \). This would give \( \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B} \). Hence proved.

Key Concepts

Difference of Angles

Understanding the difference of angles is crucial when dealing with trigonometric identities. Specifically, the identities for the sine and cosine of a difference of two angles, \(A\) and \(B\), provide a way to express complex trigonometric expressions.

• For sine, the identity is: \(\sin(A-B) = \sin A \cos B - \cos A \sin B\).
• For cosine, it is: \(\cos(A-B) = \cos A \cos B + \sin A \sin B\).

These identities are derived from the unit circle and can be used to break down trigonometric functions involving differences between angles into simpler, more manageable components.
They provide a method to transition from abstract angle manipulations to concrete algebraic expressions. Recognizing these patterns is a stepping stone toward mastering more complex trigonometric solutions.

Sine and Cosine Rules

The sine and cosine rules are foundational in trigonometry and describe relationships between the angles and sides of triangles.
When working with a difference of angles, these rules help simplify and prove various trigonometric identities.

• The sine rule expresses the relationship between the sine of angles and the lengths of their opposite sides. It's given by \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \) for a triangle with sides \( a, b, \) and \( c \) opposite angles \( A, B, \) and \( C \).
• The cosine rule, important for solving triangles, relates the cosine of an angle to the lengths of all three sides: \( c^2 = a^2 + b^2 - 2ab\cos C \).

These rules are tools that can be employed to transition from general results, such as the difference of angles identities, to more specific cases required in trigonometric proofs and applications. Understanding these can significantly ease the solving process.

Tangent Definition

Tangent is one of the fundamental functions in trigonometry and is defined as the ratio of the sine to the cosine of an angle.
This is expressed as \( \tan x = \frac{\sin x}{\cos x} \).

• When dealing with the tangent of a difference between two angles, say \( A \) and \( B \), this definition can be extended: \( \tan(A-B) = \frac{\sin(A-B)}{\cos(A-B)} \).
• This approach aligns with using the sine and cosine difference identities to express \( \tan(A-B) \) in a form that can be simplified further.

Using the definition of tangent alongside identities for the sine and cosine of angle differences allows for manipulation of tangents into forms that reveal underlying relationships.
Thus, by substituting known identities, one can simplify and evaluate expressions, leading to insightful trigonometric solutions and verifications.