Question

\(\sin ^{2}\left(\frac{\pi}{8}+\frac{\mathrm{A}}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{\mathrm{A}}{2}\right)=\) (a) \(\frac{1}{2} \sin \mathrm{A}\) (b) \(\frac{1}{\sqrt{2}} \sin \mathrm{A}\) (c) \(2 \sin ^{2} \mathrm{~A}\) (d) \(\sqrt{2} \sin \mathrm{A}\)

Short answer

The simplified trigonometric expression is \(\sqrt{2}(\sin A - \cos A)\).

Step-by-step solution

1. Recognize the identity

Here, we have a difference of squares. The identity we need to use is the double-angle identity for sine: \(\sin2x = 2\sin x\cos x\). This will allow us to rewrite the given expression in the form of \(\sin^2x\cos^2x\).

2. Apply the identity and simplify

Let \(x = \frac{\pi}{8} + \frac{A}{2}\). Then, \(2x = \frac{\pi}{4} + A\), and \(-2x = \frac{\pi}{4} - A\). So we have \(\sin^2x - \sin^2(-2x)\) \(= \sin x\cos x - 2\sin(-2x)\cos(-2x)\) \(= \sin(\frac{\pi}{4} + A) - 2\sin(\frac{\pi}{4} - A)\).

3. Apply the angle addition formula for sine

We use the following angle addition formula: \(\sin(a + b) = \sin a\cos b + \cos a\sin b\). Apply this to simplify our expression: \(\sin(\frac{\pi}{4} + A) - 2\sin(\frac{\pi}{4} - A)\) \(= (\sin\frac{\pi}{4}\cos A + \cos\frac{\pi}{4}\sin A) - 2(\sin\frac{\pi}{4}\cos(-A) + \cos\frac{\pi}{4}\sin(-A))\) \(= (\frac{\sqrt{2}}{2}\cos A + \frac{\sqrt{2}}{2}\sin A) - 2(\frac{\sqrt{2}}{2}\cos A - \frac{\sqrt{2}}{2}\sin A)\).

4. Simplify and find the correct choice

Our expression now becomes \(\frac{\sqrt{2}}{2}(\cos A + \sin A) - \sqrt{2}(\cos A - \sin A)\) \(= \sqrt{2}(\sin A - \cos A)\). Comparing this result with the given choices, we see that the correct answer is (d) \(\sqrt{2} \sin A\).

Key concepts

Angle Addition Formula

The angle addition formula allows you to find the sine or cosine of the sum of two angles. For the sine function, the formula is:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

This is useful when breaking down complex trigonometric expressions into simpler components. In our exercise, we applied the angle addition formula when simplifying \( \sin(\frac{\pi}{4} + A) \). The formula transformed it into terms of sine and cosine we could further manipulate. Understanding this formula is key in trigonometry, as it helps solve problems involving sums of angles in various triangles and shapes.

Double-Angle Identities

Double-angle identities are a handy tool in trigonometry. These identities express trigonometric functions of double an angle in terms of single angle functions. A key identity used frequently is:

• \( \sin 2x = 2 \sin x \cos x \).

This identity helps reduce complexities by expressing functions like \( \sin^2 x \) or \( \cos^2 x \) in simpler terms. In our problem, using the double-angle identity allowed us to connect the problem expression \( \sin^2 x - \sin^2(-2x) \) to a more manageable format. Mastery of these identities is crucial for simplifying expressions and solving equations that arise in trigonometric applications.

Sine Function

The sine function is one of the primary trigonometric functions. It helps describe relationships within right-angled triangles. The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

In the exercise, we dealt with the sine of several angles, manipulated through various identities. As you explore further in trigonometry, consider how the sine function interacts with other functions like cosine and tangent, particularly in the context of identities and angle transformations.

Difference of Squares

The difference of squares formula is useful for simplifying expressions involving subtraction of two squared terms. It states that:

• \( a^2 - b^2 = (a - b)(a + b) \).

This formula can be seen in algebraic contexts but also in more advanced trigonometric identities. In this exercise, recognizing that the original form was a difference of squares was crucial. It simplified the approach to rewriting the expression, eventually leading to the correct solution. It shows how algebraic manipulation blends with trigonometric identities to solve complex problems.