Problem 39 - Algebra and Trigonometry

Question

Prove the identity. $$ \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y $$

Step-by-Step Solution

Answer

The identity is proven as \( \tan y \).

Step 1: Use Sum-to-Product Identities for Sine

The identity for the sine of the sum and difference is given by:\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \] and \[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]. Using these, the expression \( \sin(x+y) - \sin(x-y) \) can be simplified to:\[ (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y) = 2 \cos x \sin y \].

Step 2: Use Sum-to-Product Identities for Cosine

The identity for the cosine of the sum and difference is given by:\[ \cos(x+y) = \cos x \cos y - \sin x \sin y \] and \[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]. Using these, the expression \( \cos(x+y) + \cos(x-y) \) can be simplified to:\[ (\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) = 2 \cos x \cos y \].

Step 3: Simplify the Expression

Substitute the results from Step 1 and Step 2 back into the given expression:\[ \frac{\sin(x+y) - \sin(x-y)}{\cos(x+y) + \cos(x-y)} = \frac{2 \cos x \sin y}{2 \cos x \cos y} \].The 2 and \( \cos x \) terms cancel, simplifying to:\[ \frac{\sin y}{\cos y} = \tan y \].

Step 4: Verify the Identity

The resultant expression \( \tan y \) matches exactly with the right-hand side of the original identity, thus confirming the identity:\[ \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y \].

Key Concepts

Understanding Sum-to-Product Identities

Sum-to-Product Identities are essential tools in trigonometry that help reduce complex expressions into more manageable forms. These identities allow us to transform sums or differences of trigonometric functions into products. This can greatly simplify calculations and proofs. For instance, when dealing with the sine of the sum and difference:

• \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
• \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)

Using these formulas, we can transform \( \sin(x+y) - \sin(x-y) \) into the product expression \( 2 \cos x \sin y \).
Similarly, for cosine, we find:

• \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
• \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)

These can be combined to simplify \( \cos(x+y) + \cos(x-y) \) to the expression \( 2 \cos x \cos y \). Understanding how to shift from sums to products is vital for simplifying and solving trigonometric problems.

Exploring Sine and Cosine Formulas

Sine and cosine are perhaps the most fundamental trigonometric functions. They describe waves and oscillations and appear in varied fields from physics to engineering. Their formulas for sum and difference are pivotal in solving more complex trigonometric problems. When working with these formulas, keep in mind:

• The sum formula for sine, \( \sin(x+y) = \sin x \cos y + \cos x \sin y \), combines the sine and cosine of individual angles.
• The difference formula for sine, \( \sin(x-y) = \sin x \cos y - \cos x \sin y \), follows a similar pattern with a sign change.

These expressions are derived from the unit circle and additional trigonometric identities. They are incredibly useful for simplifying expressions and proving identities.
The cosine formulas are analogous:

• For sums, \( \cos(x+y) = \cos x \cos y - \sin x \sin y \).
• For differences, \( \cos(x-y) = \cos x \cos y + \sin x \sin y \).

With these formulas, you can effectively manipulate trigonometric expressions to reveal deeper mathematical relationships. Sine and cosine act as the building blocks of much larger and intricate trigonometric networks.

Mastering Trigonometric Proofs

Trigonometric proofs are exercises in logical reasoning and algebraic manipulation using trigonometric identities. These proofs challenge students to bridge different concepts and arrive at a conclusion, such as demonstrating equalities or identities like the one in this exercise.
In our specific example, the identity \( \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y \) was proven through systematic application of sum-to-product identities. The key steps were:

• Simplifying the numerator \( \sin(x+y) - \sin(x-y) \) using sine formulas to get \( 2 \cos x \sin y \).
• Simplifying the denominator \( \cos(x+y) + \cos(x-y) \) using cosine formulas to obtain \( 2 \cos x \cos y \).
• Reducing the final expression by canceling out common factors, leading to \( \frac{\sin y}{\cos y} \), which equals \( \tan y \).

Through practice, you can enhance your skills in recognizing which identities to use and how to manipulate expressions effectively. These proofs are not only a mental exercise but also a way to solidify understanding of trigonometric relationships.