Problem 62 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\cos x \sin y=\frac{1}{2}[\sin (x+y)-\sin (x-y)]$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity: $$\cos x \sin y = \frac{1}{2}[\sin (x+y) - \sin (x-y)]$$ Answer: We proved the given trigonometric identity by using the sine addition and subtraction formulas. We substituted the expressions for \(\sin (x+y)\) and \(\sin (x-y)\), simplified the right-hand side of the identity, and verified that it matches the left-hand side. Therefore, the given identity is true: $$\cos x \sin y = \frac{1}{2}[\sin (x+y) - \sin (x-y)]$$

Step 1: Write the given identity

We have to prove the following identity: $$\cos x \sin y = \frac{1}{2}[\sin (x+y) - \sin (x-y)]$$

Step 2: Apply the sine addition formula to \(\sin (x+y)\)

We can rewrite \(\sin (x+y)\) using the sine addition formula: $$\sin (x+y) = \sin x \cos y + \cos x \sin y$$

Step 3: Apply the sine subtraction formula to \(\sin (x-y)\)

We can rewrite \(\sin (x-y)\) using the sine subtraction formula: $$\sin (x-y) = \sin x \cos y - \cos x \sin y$$

Step 4: Substitute the expressions from Steps 2 and 3 into the right-hand side of the identity

Replace \(\sin (x+y)\) and \(\sin (x-y)\) with their respective expressions from Steps 2 and 3: $$\frac{1}{2}[\sin (x+y) - \sin (x-y)] = \frac{1}{2}[(\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y)]$$

Step 5: Simplify the right-hand side of the identity

Distribute the \(1/2\) and remove the parentheses: $$\frac{1}{2}[(\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y)] = \frac{1}{2} \sin x \cos y + \frac{1}{2} \cos x \sin y - \frac{1}{2} \sin x \cos y + \frac{1}{2} \cos x \sin y$$ Combine the like terms: $$\frac{1}{2} \cos x \sin y + \frac{1}{2} \cos x \sin y = \cos x \sin y$$

Step 6: Compare the left-hand side and the right-hand side

The simplified version of the right-hand side matches the left-hand side of the identity: $$\cos x \sin y = \cos x \sin y$$

Step 7: Conclusion

We have proved the given trigonometric identity: $$\cos x \sin y = \frac{1}{2}[\sin (x+y) - \sin (x-y)]$$

Key Concepts

Sine Addition Formula

The Sine Addition Formula is a key trigonometric identity, and it's quite useful when you need to find the sine of a sum of two angles. This formula states:

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

This means that to find the sine of two angles added together, you need to multiply the sine of the first angle by the cosine of the second, add to it the product of the cosine of the first angle and the sine of the second. This formula helps in transforming complex trigonometric expressions into simpler ones by breaking them down into known parts.
For example, if you need to find \(\sin(45^\circ + 30^\circ)\), you can use \(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\) and \(\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}\). Then you find the result by calculating:

• \(\sin(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

The Sine Addition Formula is crucial for solving problems in trigonometry where angle transformations are needed.

Sine Subtraction Formula

The Sine Subtraction Formula is similar to its addition counterpart and it helps to find the sine of the difference of two angles. The formula is given by:

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

Here, to calculate the sine of one angle minus another, you subtract the product of the cosine of the first angle and the sine of the second from the product of the sine of the first angle and the cosine of the second. This comes in handy when simplifying expressions involving angle differences.
Consider you want to compute \(\sin(60^\circ - 30^\circ)\), using the known values \(\sin 60^\circ = \frac{\sqrt{3}}{2}, \cos 60^\circ = \frac{1}{2}\),\(\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}\), the formula simplifies the expression:

• \(\sin(60^\circ - 30^\circ) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}\)

The Sine Subtraction Formula is very useful for proofs and trigonometric calculations involving the difference of angles.

Trigonometry Proof

Trigonometry proofs involve demonstrating that two different forms of a trigonometric expression are identical. These proofs often require the application of known identities such as addition, subtraction, or Pythagorean identities, like the proof given in the exercise.In the given problem, the goal was to prove that \(\cos x \sin y = \frac{1}{2}[\sin(x+y) - \sin(x-y)]\). To do this, both the Sine Addition and Sine Subtraction Formulas were used to transform the initial identity. By expressing \(\sin(x+y)\) and \(\sin(x-y)\) as combinations of sine and cosine terms, we simplify the expression.

• Step 1: Identify the identity to be proven and separate complex expressions using known formulas.
• Step 2: Substitute these formulas back into the expression.
• Step 3: Simplify the resulting expression by combining like terms and arranging similar patterns.

For the identity \(\cos x \sin y = \frac{1}{2}[\sin(x+y) - \sin(x-y)]\), upon simplification, the terms combined perfectly to reproduce the left side of the identity, proving the statement. This approach illustrates how algebraic manipulation and formula application are key strategies in solving trigonometric proofs.