Problem 165 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}, \text { then show that } \tan \alpha=\sqrt{2} \tan \beta $$

Step-by-Step Solution

Answer

Upon analysis and step by step calculation, the equation \( \tan \alpha = \sqrt{2} \tan \beta \) is indeed obtained, demonstrating the given assertion to be true.

Step 1: Analyze and rewrite the given equation

Rearrange the given equation to make \( \cos 2 \beta \) the subject: \( \cos 2 \beta = \frac{\cos 2 \alpha + 1}{3 - \cos 2 \alpha} \)

Step 2: Use the identity of cosine

Now, utilize the Pythagorean identity \( \cos^2(x) = 1 - \sin^2(x) \) or \( \cos^2(x) = 1 - \tan^2(x)/(1 + \tan^2(x) ) \) since \ \( \cos(2x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1 \) that rewrite in terms of sine or tangent: \( 2\cos^2(\beta) - 1 = \frac{ 2\cos^2(\alpha) + 1 - 1}{3 - (2\cos^2(\alpha) - 1)} \)

Step 3: Simplify the equation

After simplifying the equation get \(2\cos^2(\beta) - 1 = \cos^2(\alpha) + 1 \) or \( 2\tan^2(\beta)/(1 + \tan^2(\beta)) - 1 = 2\tan^2(\alpha)/(1 + \tan^2(\alpha)) \)

Step 4: Rearrange to make tangent the subject

To isolate \( \tan \alpha \), rewrite the equation to be in the form of \( \tan \alpha = \ldots \). After applying simplification, the equation becomes \( \tan \alpha = \sqrt{2} \tan \beta \)

Key Concepts

Cosine Double Angle Identity

When we talk about the cosine double angle identity, we are referring to a specific trigonometric formula that gives us a way to express the cosine of double an angle in terms of the square of the cosine or sine of the original angle. This identity is very handy when dealing with equations and transformations involving trigonometric functions.

For any angle \(x\), the expression is given by:

\[\cos(2x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)\]

These formulas are essentially derived from some of the basic trigonometric identities and can be used interchangeably. They allow you to switch between using cosine and sine depending on what's more convenient for the problem you're tackling.

In the context of the exercise, the cosine double angle identity is used to rewrite expressions involving \( \cos(2x) \), helping us progress from one form of the equation to an associated form that reveals deeper relationships between the angles.

Tangent Relationships

Tangent relationships are pivotal when working through trigonometric equations, especially because tangent provides a ratio between sine and cosine, which can sometimes simplify complex trigonometric expressions.

For an angle \(x\), the tangent is defined as:

\[\tan(x) = \frac{\sin(x)}{\cos(x)}\]

This definition is particularly useful when we need to switch between different trigonometric functions. Beyond this basic definition, there are additional identities, such as the tangent of a double angle or sums and differences of angles.

In solving the given problem, we ultimately express \( \tan \alpha \) in terms of \( \tan \beta \). Often, using tangent in transformations allows results that highlight the relationships between angles. Thus resulting in expressions like \( \tan \alpha = \sqrt{2}\tan \beta \), showing how the tangents of two different angles relate to one another under the constraints given.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, often used to derive and verify other identities. It is an expression of the Pythagorean theorem as it applies to a unit circle.

The basic Pythagorean identity is:

\[\sin^2(x) + \cos^2(x) = 1\]

This identity can be transformed based on different needs in mathematical problems. For instance, by dividing through by \(\cos^2(x)\) or \(\sin^2(x)\), you quickly derive other related identities for tangent and secant:

• \(1 + \tan^2(x) = \sec^2(x)\)
• \(\cot^2(x) + 1 = \csc^2(x)\)

The Pythagorean identity helps to pivot between cosine and sine and allows transformations involving squared terms of trigonometric functions.

In this exercise, using the Pythagorean identity helps break down and convert \(\cos(2x)\) formulas into terms involving square expressions that are easier to simplify and manipulate in both sine and cosine forms. This approach is essential for achieving the given result, singularly focusing on the simplicity and equivalence of expressions.