Problem 60 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin ^{2}\left(\frac{\pi}{8}+\frac{A}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{A}{2}\right)=\frac{1}{\sqrt{2}} \sin A $$

Step-by-Step Solution

Answer

In order to solve the given equation, we first convert the equation into a form involving \(\sin A\) by applying sine double angle formulas. After simplifying the expressions and canceling common terms, we find that the equation becomes \(0 = \frac{1}{\sqrt{2}}\sin A\). For this equation to hold true, we need \(\sin A = 0\), which is valid when \(A = n\pi\) where n is an integer. Thus, the solution for \(A\) is any integer multiple of \(\pi\).

Step 1: Apply the sine double angle formula

Apply the sine double angle formula for both the terms in the equation: \[ \sin(2\alpha) = 2\sin\alpha\cos\alpha \] So, we have: \[ \sin^2\left(\frac{\pi}{8}+\frac{A}{2}\right) = \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{A}{2}\right)\right)^2 \\ \sin^2\left(\frac{\pi}{8}-\frac{A}{2}\right) = \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(-\frac{A}{2}\right)\right)^2 \]

Step 2: Use the properties of cosine to simplify the equation

Use cosine property (\(\cos(-\theta) = \cos(\theta)\)) to simplify the equation: \[ \sin^2\left(\frac{\pi}{8}+\frac{A}{2}\right) = \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{A}{2}\right)\right)^2 \\ \sin^2\left(\frac{\pi}{8}-\frac{A}{2}\right) = \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{A}{2}\right)\right)^2 \]

Step 3: Substitute the expressions back into the original equation

Replace the sine terms with their simplified values: \[ \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{A}{2}\right)\right)^2 - \left(2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{A}{2}\right)\right)^2 = \frac{1}{\sqrt{2}}\sin A \]

Step 4: Observe the subtraction and cancel common terms

The left side of the equation has subtraction of same expressions, which leads to a zero on the left side. Thus, the equation becomes: \[ 0 = \frac{1}{\sqrt{2}}\sin A \]

Step 5: Evaluate \(A\)

In order for the equation to hold true, we need to find the value of \(A\) such that \(\sin A\) evaluates to zero: \[ \sin A = 0 \] This is true for \(A = n\pi\) where n is an integer. Therefore, the solution for \(A\) is any integer multiple of \(\pi\).

Key Concepts

Sine Double Angle Formula

The sine double angle formula is a key identity in trigonometry that allows you to express the sine of double an angle in terms of sine and cosine of the original angle. This formula is depicted as:

• \[\sin(2\alpha) = 2\sin\alpha\cos\alpha\]

This can be particularly useful when solving trigonometric equations or transforming expressions into simpler forms. In the original exercise, the usage of this formula helps to rewrite complicated sine terms into a standard form that can be easily manipulated. For example, expressing \(\sin^2\left(\frac{\pi}{8}+\frac{A}{2}\right)\) using double angle identities simplifies the task by breaking down what could be a complex trigonometric form into something familiar and smaller parts. By understanding how angles and their trigonometric functions interact, such methods reduce complexity and enable easier calculation or solution finding.

Cosine Properties

Cosine properties serve as essential tools for simplifying various trigonometric expressions and equations. A key property employed in this exercise is that the cosine of a negative angle is equivalent to the cosine of the positive angle:

• \[\cos(-\theta) = \cos(\theta)\]

This identity comes from the fact that cosine is an even function, meaning it has symmetry about the y-axis. This symmetry can be useful in solving problems where you need to simplify expressions that include cosines of negative angles. In our original step by step solution, applying this property directly transforms the expression, making it more manageable and insightful. Incorporating symmetry properties like this allows one to see patterns or simplifications that aren't immediately obvious, providing a pathway to unravel complex trigonometric solutions.

Trigonometric Equations

Solving trigonometric equations involves finding angles (or values) that satisfy a given equation involving trigonometric functions. These equations often use identities and properties to reduce and simplify the initial expression. In the solution to the exercise, after applying trigonometric identities such as the sine double angle and cosine properties, the simplified equation helps determine the values of \(A\) that satisfy:

• \[0 = \frac{1}{\sqrt{2}}\sin A\]

By observing that \(\sin A = 0\) solves the equation, you can conclude that \(A\) must be an integer multiple of \(\pi\). This arises from the fact that sine is zero at integer multiples of \(\pi\), translating to possible values for \(A\) such as \(0, \pi, 2\pi\), and so on. Understanding how to leverage trigonometric properties and identities simplifies the process of finding solutions to these equations and is crucial for mastering trigonometry.