Understanding Product to Sum Formulae in Trigonometry

Product to sum formulae are standard trigonometric identities that express the product of sine and cosine functions as algebraic sums or differences of trigonometric terms in the same variables.

Formal Expressions of the Product to Sum Formulae in Trigonometry

Four canonical formulae convert products of $\sin$ and $\cos$ functions into sums or differences using angle addition and subtraction notations.

For any real numbers $A$ and $B$, the identities are given by:

$\sin A \sin B = \dfrac{1}{2}\left[\cos(A - B) - \cos(A + B)\right]$

$\cos A \cos B = \dfrac{1}{2}\left[\cos(A + B) + \cos(A - B)\right]$

$\sin A \cos B = \dfrac{1}{2}\left[\sin(A + B) + \sin(A - B)\right]$

$\cos A \sin B = \dfrac{1}{2}\left[\sin(A + B) - \sin(A - B)\right]$

These identities facilitate the direct algebraic manipulation of trigonometric products, essential in simplification and integration.

Derivation of Product to Sum Relations from Angle Addition Identities

The derivation proceeds systematically from the fundamental sum and difference formulas for sine and cosine, namely:

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

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

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

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

By algebraic addition and subtraction of these four formulas, the product to sum formulae can be systematically obtained.

Adding $\sin(A+B)$ and $\sin(A-B)$ yields $2\sin A \cos B$; solving for $\sin A \cos B$ produces the corresponding product to sum identity. Analogous calculations obtain the other three relations.

Transformation Techniques Using Product to Sum Relations

Frequently, the product to sum identities are essential for evaluating integrals or transforming algebraic expressions in which trigonometric products appear as non-standard angles.

Such transformations are a direct consequence of applying the above identities according to the functional structure of the given expression.

Illustrative Problems Involving Product to Sum Formulae

Example 1. Evaluate $\sin 75^\circ \sin 15^\circ$ without using tables.

$\sin 75^\circ \sin 15^\circ = \dfrac{1}{2}[\cos(75^\circ - 15^\circ) - \cos(75^\circ + 15^\circ)]$

$= \dfrac{1}{2}[\cos 60^\circ - \cos 90^\circ]$

$= \dfrac{1}{2}\left[\dfrac{1}{2} - 0\right] = \dfrac{1}{4}$

Example 2. Express $2\cos 5x \sin 2x$ as a sum or difference of trigonometric functions.

$\cos 5x \sin 2x = \dfrac{1}{2}[\sin(5x + 2x) - \sin(5x - 2x)]$

$= \dfrac{1}{2}[\sin 7x - \sin 3x]$

Therefore, $2\cos 5x \sin 2x = \sin 7x - \sin 3x$

Example 3. Find $\displaystyle\int \sin 3x \cos 4x \, dx$.

$\sin 3x \cos 4x = \dfrac{1}{2}[\sin (3x + 4x) + \sin (3x - 4x)]$

$= \dfrac{1}{2}[\sin 7x + \sin(-x)] = \dfrac{1}{2}[\sin 7x - \sin x]$

$\displaystyle\int \sin 3x \cos 4x \, dx = \dfrac{1}{2}\int (\sin 7x - \sin x)\, dx$

$= \dfrac{1}{2} \left(-\dfrac{\cos 7x}{7} + \cos x\right) + C$

Example 4. Simplify $\cos 15^\circ \sin 45^\circ$ using the product to sum identity.

$\cos 15^\circ \sin 45^\circ = \dfrac{1}{2}[\sin(15^\circ + 45^\circ) - \sin(15^\circ - 45^\circ)]$

$= \dfrac{1}{2}[\sin 60^\circ - \sin(-30^\circ)] = \dfrac{1}{2}[\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}] = \dfrac{\sqrt{3} + 1}{4}$

Sum To Product Formulae are the converse identities and are useful in reversing this transformation.

Typical JEE Patterns Assessing Product to Sum Formulae

• Evaluation of products of trigonometric values
• Simplification of algebraic trigonometric expressions
• Integration requiring transformation of products to sums
• Proof-based identities involving non-standard angles
• Conversion tasks between compound and simple angles

Careful substitution and algebraic discipline are crucial for avoiding sign errors in application of these identities during examinations.

For more details on related transformations, see Trigonometric Ratios Of Compound Angles and other core formulae.