Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta}=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta} $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\tan(\alpha + \beta)\).

Step 1: Recall Trigonometric Identities

To solve this problem, let's recall key trigonometric identities:1) \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)2) \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)3) \(\tan A = \frac{\sin A}{\cos A}\) and \(\tan B = \frac{\sin B}{\cos B}\).

Step 2: Apply Sum of Angles Identities to the Left Side

Notice that the left side of the equation can be seen as \( \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} \).Apply the sum of angles identity formula:\[\sin \alpha \cos \beta + \cos \alpha \sin \beta = \sin(\alpha + \beta)\]\[\cos \alpha \cos \beta - \sin \alpha \sin \beta = \cos(\alpha + \beta)\].Thus, the left side simplifies to \( \frac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)}\), which equals \(\tan(\alpha + \beta)\).

Step 3: Simplify the Right Side Using Tan Addition Formula

The right side of the equation is \(\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\), which is the tangent of a sum identity for \(\tan(\alpha + \beta)\).

Step 4: Verify Both Sides are Equal

By transformation, both sides have become \(\tan(\alpha + \beta)\). Therefore, both sides are equal, verifying the identity.

Key Concepts

Sum of Angles Identity

The sum of angles identity helps us find the sine or cosine of the sum of two angles. This identity is handy when simplifying complex trigonometric expressions or verifying identities, like in our original exercise. For sine, the identity is:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

This formula allows us to express the sine of a combined angle using the sines and cosines of the individual angles.
Similarly, for cosine, the identity is:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

These identities are foundational and greatly simplify working with trigonometric functions.
In our step-by-step solution, these identities were crucial in transforming the complex expression on the left side into a more manageable form, leading to the simplified \( \tan(\alpha + \beta) \).

Tangent Addition Formula

The tangent addition formula is another essential trigonometric identity which helps express the tangent of a sum of two angles. It is given by:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

This identity simplifies the process of calculating the tangent for a sum of angles, especially when dealing with expressions that involve multiple tangent terms.
In our original exercise, the right side of the equation \( \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \) directly applies this formula.
This step was crucial in verifying our identity as both sides eventually become the same trigonometric expression, demonstrating their equivalence.

Verifying Trigonometric Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation represent the same value through substitution and transformation using known identities. It's like solving a mathematical puzzle where each piece is a known trigonometric rule.
Start by simplifying either or both sides using identities like the sum of angles or the tangent addition formula. Sometimes one side is already simplified, and you just need to transform the other side to match it.
In our exercise, both sides were transformed to equal \( \tan(\alpha + \beta) \), confirming they are indeed the same.

• Use identities to simplify each side.
• Translate complex terms into simpler or equivalent expressions.
• Conclude when both sides match perfectly.

Mastering this process improves problem-solving skills and deepens understanding of trigonometric relationships.