📐 Level 2 - Topic 12: Product-to-Sum & Sum-to-Product Identities - Transforming Products & Sums 🔄🔗

1) Introduction to Product-to-Sum and Sum-to-Product Identities - Transformations 🔄

Welcome to **Level 2 - Topic 12: Product-to-Sum and Sum-to-Product Identities**. In this topic, we will explore identities that transform products of trigonometric functions into sums, and conversely, sums of trigonometric functions into products. These identities are powerful tools for simplifying expressions, solving equations, and are particularly useful in areas like signal processing and physics where waveform analysis is crucial.

What are Product-to-Sum and Sum-to-Product Identities?

Product-to-Sum Identities: Formulas that convert a product of two trigonometric functions (like \( \sin(A)\cos(B) \), \( \cos(A)\cos(B) \), \( \sin(A)\sin(B) \)) into a sum or difference of trigonometric functions.
Sum-to-Product Identities: Formulas that convert a sum or difference of two trigonometric functions (like \( \sin(x) + \sin(y) \), \( \cos(x) - \cos(y) \), etc.) into a product of trigonometric functions.
These identities are derived from the angle addition and subtraction formulas we covered in Topic 8.

Why are these Identities Important?

Simplification: Convert complex products or sums into simpler forms for algebraic manipulation, integration, differentiation, etc.
Solving Equations: Transform equations into forms that are easier to solve by factoring or using other techniques.
Harmonic Analysis: In signal processing, these identities are used to analyze and synthesize waveforms by converting between product and sum representations of frequencies.
Physics and Engineering: Applications in wave phenomena, optics, acoustics, and more, where understanding superposition and interference patterns is vital.

In this topic, we will derive and apply both sets of identities. We will see how products can be expressed as sums, and how sums can be expressed as products, enhancing our trigonometric manipulation skills. Let's start transforming! 🔄🔗

2) Product-to-Sum Identities - Products to Sums ➕

Let's first derive the product-to-sum identities. These will express products of sines and cosines as sums or differences.

Definition: Product-to-Sum Identities

The product-to-sum identities are:

1. \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \)

2. \( \cos(A)\sin(B) = \frac{1}{2}[\sin(A + B) - \sin(A - B)] \)

3. \( \cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \)

4. \( \sin(A)\sin(B) = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \)

Derivation from Angle Sum and Difference Identities (Topic 8):

Identity 1 Derivation \( \sin(A)\cos(B) \):
Start with sum and difference formulas for sine:

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

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

Add these two equations:

\( \sin(A + B) + \sin(A - B) = 2\sin(A)\cos(B) \)

Divide by 2 to isolate \( \sin(A)\cos(B) \):

\( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \)
Identity 2 Derivation \( \cos(A)\sin(B) \):
Start with the same sum and difference formulas for sine. Subtract the second equation from the first:

\( \sin(A + B) - \sin(A - B) = 2\cos(A)\sin(B) \)

Divide by 2 to isolate \( \cos(A)\sin(B) \):

\( \cos(A)\sin(B) = \frac{1}{2}[\sin(A + B) - \sin(A - B)] \)
Identity 3 Derivation \( \cos(A)\cos(B) \):
Start with sum and difference formulas for cosine:

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

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

Add these two equations:

\( \cos(A + B) + \cos(A - B) = 2\cos(A)\cos(B) \)

Divide by 2 to isolate \( \cos(A)\cos(B) \):

\( \cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \)
Identity 4 Derivation \( \sin(A)\sin(B) \):
Start with the same sum and difference formulas for cosine. Subtract the first equation from the second:

\( \cos(A - B) - \cos(A + B) = 2\sin(A)\sin(B) \)

Divide by 2 to isolate \( \sin(A)\sin(B) \):

\( \sin(A)\sin(B) = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \)

Example 1: Using Product-to-Sum Identities

Express the product \( \sin(4x)\cos(2x) \) as a sum of trigonometric functions.

Identify the appropriate product-to-sum identity: We have a product of sine and cosine, so use Identity 1: \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \).
Set \( A = 4x \) and \( B = 2x \): Substitute these into the identity.

\( \sin(4x)\cos(2x) = \frac{1}{2}[\sin(4x + 2x) + \sin(4x - 2x)] \)
Simplify:

\( \sin(4x)\cos(2x) = \frac{1}{2}[\sin(6x) + \sin(2x)] \)

Solution: \( \sin(4x)\cos(2x) = \frac{1}{2}\sin(6x) + \frac{1}{2}\sin(2x) \). The product is now expressed as a sum of sines.

Example 2: Evaluating Expression Using Product-to-Sum Identity

Evaluate \( \cos(75^\circ)\cos(15^\circ) \) using a product-to-sum identity.

Identify the appropriate identity: Product of cosines, so use Identity 3: \( \cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \).
Set \( A = 75^\circ \) and \( B = 15^\circ \): Substitute these into the identity.

\( \cos(75^\circ)\cos(15^\circ) = \frac{1}{2}[\cos(75^\circ + 15^\circ) + \cos(75^\circ - 15^\circ)] \)
Simplify and evaluate:

\( \cos(75^\circ)\cos(15^\circ) = \frac{1}{2}[\cos(90^\circ) + \cos(60^\circ)] \)

\( \cos(75^\circ)\cos(15^\circ) = \frac{1}{2}[0 + \frac{1}{2}] = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \)

Solution: \( \cos(75^\circ)\cos(15^\circ) = \frac{1}{4} \).

3) Sum-to-Product Identities - Sums to Products ➖

Now, let's derive the sum-to-product identities, which do the reverse – transform sums and differences into products.

Definition: Sum-to-Product Identities

The sum-to-product identities are:

1. \( \sin(x) + \sin(y) = 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \)

2. \( \sin(x) - \sin(y) = 2\cos\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right) \)

3. \( \cos(x) + \cos(y) = 2\cos\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \)

4. \( \cos(x) - \cos(y) = -2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right) \) or \( = 2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{y - x}{2}\right) \)

Derivation from Product-to-Sum Identities (by variable substitution):

Identity 1 Derivation \( \sin(x) + \sin(y) \):
Start with product-to-sum identity 1: \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \). Let \( x = A + B \) and \( y = A - B \). Solve for \( A \) and \( B \) in terms of \( x \) and \( y \): Adding the two equations gives \( x + y = 2A \Rightarrow A = \frac{x + y}{2} \). Subtracting the second from the first gives \( x - y = 2B \Rightarrow B = \frac{x - y}{2} \).
Substitute \( A = \frac{x + y}{2} \) and \( B = \frac{x - y}{2} \) into product-to-sum identity 1.

\( \sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) = \frac{1}{2}[\sin((\frac{x + y}{2}) + (\frac{x - y}{2})) + \sin((\frac{x + y}{2}) - (\frac{x - y}{2}))] \)

\( \sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) = \frac{1}{2}[\sin(\frac{2x}{2}) + \sin(\frac{2y}{2})] = \frac{1}{2}[\sin(x) + \sin(y)] \)

Multiply by 2 to isolate \( \sin(x) + \sin(y) \):

\( \sin(x) + \sin(y) = 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \)
Identity 2, 3, 4 Derivations: Follow a similar process starting from product-to-sum identities 2, 3, and 4, using the same substitutions \( x = A + B \) and \( y = A - B \), and solve for \( A \) and \( B \). Then substitute back into the respective product-to-sum identities and rearrange to get the sum-to-product forms.

Example 3: Using Sum-to-Product Identity

Express the sum \( \sin(5x) + \sin(3x) \) as a product of trigonometric functions.

Identify the appropriate sum-to-product identity: Sum of sines, use Identity 1: \( \sin(x) + \sin(y) = 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \).
Set \( x = 5x \) and \( y = 3x \): Substitute into the identity.

\( \sin(5x) + \sin(3x) = 2\sin\left(\frac{5x + 3x}{2}\right)\cos\left(\frac{5x - 3x}{2}\right) \)
Simplify:

\( \sin(5x) + \sin(3x) = 2\sin\left(\frac{8x}{2}\right)\cos\left(\frac{2x}{2}\right) = 2\sin(4x)\cos(x) \)

Solution: \( \sin(5x) + \sin(3x) = 2\sin(4x)\cos(x) \). The sum is now expressed as a product.

Example 4: Evaluating Expression Using Sum-to-Product Identity

Evaluate \( \cos(105^\circ) - \cos(15^\circ) \) using a sum-to-product identity.

Identify the appropriate identity: Difference of cosines, use Identity 4: \( \cos(x) - \cos(y) = -2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right) \).
Set \( x = 105^\circ \) and \( y = 15^\circ \): Substitute into the identity.

\( \cos(105^\circ) - \cos(15^\circ) = -2\sin\left(\frac{105^\circ + 15^\circ}{2}\right)\sin\left(\frac{105^\circ - 15^\circ}{2}\right) \)
Simplify and evaluate:

\( \cos(105^\circ) - \cos(15^\circ) = -2\sin\left(\frac{120^\circ}{2}\right)\sin\left(\frac{90^\circ}{2}\right) = -2\sin(60^\circ)\sin(45^\circ) \)

\( \cos(105^\circ) - \cos(15^\circ) = -2 \cdot \left(\frac{\sqrt{3}}{2}\right) \cdot \left(\frac{\sqrt{2}}{2}\right) = -\frac{2\sqrt{6}}{4} = -\frac{\sqrt{6}}{2} \)

Solution: \( \cos(105^\circ) - \cos(15^\circ) = -\frac{\sqrt{6}}{2} \).

4) Verifying Identities using Product-to-Sum and Sum-to-Product Identities 🚀

Example 5: Verifying Identity using Sum-to-Product Identity

Verify the identity: \( \frac{\sin(3x) + \sin(x)}{\cos(3x) + \cos(x)} = \tan(2x) \).

Start with the left side (LS): \( \text{LS} = \frac{\sin(3x) + \sin(x)}{\cos(3x) + \cos(x)} \).
Apply sum-to-product identities to numerator and denominator:
For numerator, use Identity 1 for sum of sines: \( \sin(3x) + \sin(x) = 2\sin\left(\frac{3x + x}{2}\right)\cos\left(\frac{3x - x}{2}\right) = 2\sin(2x)\cos(x) \).
For denominator, use Identity 3 for sum of cosines: \( \cos(3x) + \cos(x) = 2\cos\left(\frac{3x + x}{2}\right)\cos\left(\frac{3x - x}{2}\right) = 2\cos(2x)\cos(x) \).

\( \text{LS} = \frac{2\sin(2x)\cos(x)}{2\cos(2x)\cos(x)} \)
Simplify by cancelling common factors: Cancel \( 2\cos(x) \) from numerator and denominator (assuming \( \cos(x) \ne 0 \)).

\( \text{LS} = \frac{\sin(2x)}{\cos(2x)} \)
Use quotient identity for tangent: \( \frac{\sin(2x)}{\cos(2x)} = \tan(2x) \).

\( \text{LS} = \tan(2x) \)
Compare with the right side (RS): \( \text{RS} = \tan(2x) \).
Conclusion: Since \( \text{LS} = \text{RS} \), the identity \( \frac{\sin(3x) + \sin(x)}{\cos(3x) + \cos(x)} = \tan(2x) \) is verified.

5) Solving Equations using Sum-to-Product Identities 🚀

Example 6: Solving Equation using Sum-to-Product Identity

Solve the equation \( \sin(3x) + \sin(x) = 0 \) for principal solutions in \( [0, 2\pi) \).

Apply sum-to-product identity to the left side: Use Identity 1 for sum of sines: \( \sin(3x) + \sin(x) = 2\sin\left(\frac{3x + x}{2}\right)\cos\left(\frac{3x - x}{2}\right) = 2\sin(2x)\cos(x) \).

Equation becomes: \( 2\sin(2x)\cos(x) = 0 \)
Set each factor to zero: Product is zero if either factor is zero.

\( 2\sin(2x) = 0 \) or \( \cos(x) = 0 \)
Solve \( \sin(2x) = 0 \): For \( \sin(\theta) = 0 \), \( \theta = n\pi \) where \( n \) is an integer. Here \( \theta = 2x \). So, \( 2x = n\pi \Rightarrow x = \frac{n\pi}{2} \). Principal solutions in \( [0, 2\pi) \) are for \( n = 0, 1, 2, 3 \), giving \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
Solve \( \cos(x) = 0 \): For \( \cos(x) = 0 \), \( x = \frac{\pi}{2} + m\pi \) where \( m \) is an integer. Principal solutions in \( [0, 2\pi) \) are for \( m = 0, 1 \), giving \( x = \frac{\pi}{2}, \frac{3\pi}{2} \).
Combine solutions and remove duplicates: Solutions from \( \sin(2x) = 0 \) are \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). Solutions from \( \cos(x) = 0 \) are \( x = \frac{\pi}{2}, \frac{3\pi}{2} \). The combined unique solutions are \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).

Solution: Principal solutions are \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).

6) Practice Questions 🎯

6.1 Fundamental – Using Product-to-Sum and Sum-to-Product Identities for Calculation and Simplification

1. Express \( \sin(3x)\sin(2x) \) as a sum or difference of cosines.

2. Express \( \cos(5x)\sin(x) \) as a sum or difference of sines.

3. Evaluate \( \sin(105^\circ)\sin(15^\circ) \).

4. Evaluate \( \cos(165^\circ)\cos(75^\circ) \).

5. Express \( \sin(7x) + \sin(x) \) as a product.

6. Express \( \cos(4x) - \cos(2x) \) as a product.

7. Evaluate \( \sin(75^\circ) + \sin(15^\circ) \).

8. Evaluate \( \cos(105^\circ) + \cos(15^\circ) \).

9. Simplify \( \frac{\sin(5x) - \sin(x)}{\cos(5x) + \cos(x)} \).

10. Simplify \( \frac{\cos(3x) - \cos(x)}{\sin(3x) + \sin(x)} \).

11. Rewrite \( 2\sin(3\theta)\cos(\theta) \) as a sum.

12. Rewrite \( 2\cos(5\theta)\cos(3\theta) \) as a sum.

6.2 Challenging – Advanced Verification and Equation Solving 💪🚀

1. Verify the identity: \( \frac{\sin(A) + \sin(B)}{\sin(A) - \sin(B)} = \tan\left(\frac{A + B}{2}\right)\cot\left(\frac{A - B}{2}\right) \).

2. Verify the identity: \( \frac{\cos(x) + \cos(3x)}{\sin(3x) - \sin(x)} = \cot(x) \).

3. Verify the identity: \( \sin(x) + \sin(2x) + \sin(3x) = \sin(2x)(1 + 2\cos(x)) \) (Hint: Group \( \sin(x) + \sin(3x) \) and use sum-to-product).

4. Solve the equation: \( \cos(3x) + \cos(x) = 0 \) for principal solutions in \( [0, 2\pi) \).

5. Solve the equation: \( \sin(5x) - \sin(3x) = \cos(4x) \) for principal solutions in \( [0, 2\pi) \) (Hint: Use sum-to-product and then consider factoring).

6. (Conceptual) Discuss the role of product-to-sum and sum-to-product identities in simplifying waveform analysis in signal processing or physics. Give a conceptual example.

7. (Challenging Simplification) Simplify \( \frac{\sin(x) + \sin(3x) + \sin(5x) + \sin(7x)}{\cos(x) + \cos(3x) + \cos(5x) + \cos(7x)} \) (Hint: Group terms pairwise and use sum-to-product).

7) Summary - Product-to-Sum & Sum-to-Product Identities - Transformations 🎉

Product-to-Sum Identities: Transform products into sums:
- \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \)
- \( \cos(A)\sin(B) = \frac{1}{2}[\sin(A + B) - \sin(A - B)] \)
- \( \cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \)
- \( \sin(A)\sin(B) = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \)
Sum-to-Product Identities: Transform sums into products:
- \( \sin(x) + \sin(y) = 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \)
- \( \sin(x) - \sin(y) = 2\cos\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right) \)
- \( \cos(x) + \cos(y) = 2\cos\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right) \)
- \( \cos(x) - \cos(y) = -2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right) \)
Derivation: Product-to-Sum from angle sum/difference formulas; Sum-to-Product via variable substitution in Product-to-Sum.
Applications: Simplification, equation solving, evaluating expressions, signal processing, harmonic analysis.
Techniques: Recognize product or sum forms, choose appropriate identity, apply identities to transform, algebraic simplification, factorization for equations.

Congratulations! You have now mastered Product-to-Sum and Sum-to-Product Identities. These identities provide powerful transformations between products and sums of trigonometric functions, greatly expanding your ability to manipulate and simplify trigonometric expressions. You've also seen how these are applied in solving equations and evaluating expressions. As you advance, you will find these identities essential in various areas of mathematics, physics, and engineering. Keep practicing, and you'll expertly transform trigonometric forms to solve complex problems! 🔄🔗💪📐🌟