Problem 259 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \cos \alpha+\cos \beta=0=\sin \alpha+\sin \beta, \text { then show that } \cos 2 \alpha+\cos 2 \beta=-2 \cos (\alpha+\beta) \text { . } $$

Step-by-Step Solution

Answer

The identity \( \cos 2 \alpha+\cos 2 \beta = -2 \cos(\alpha+\beta) \) is valid given the conditions \( \cos \alpha+\cos \beta=0 \) and \( \sin \alpha+\sin \beta=0 \).

Step 1: Given Condition

We start with the given conditions: \( \cos \alpha+\cos \beta=0 \) and \( \sin \alpha+\sin \beta=0 \). The second equation indicates that \(\alpha\) and \(\beta\) are supplementary angles. From this, we know \( \cos \alpha = -\cos \beta \) and \( \sin \alpha = -\sin \beta \)

Step 2: Expression for Cos 2α and Cos 2β

We can proceed by re-writing the expressions \( \cos 2\alpha \) and \( \cos 2\beta \) using double angle formulae: \( \cos2\alpha = 2\cos^2\alpha - 1 \) and \( \cos2\beta = 2\cos^2\beta - 1 \)

Step 3: Substitute Conditions

Let's substitute \( \cos \beta = -\cos \alpha \) into the two equations derived in the previous step: \( \cos2\alpha = 2\cos^2\alpha - 1 \) and \( \cos2\beta = 2\cos^2(-\cos\alpha) - 1 \). Simplifying these we get: \( \cos2\alpha = 2\cos^2\alpha - 1 \) and \( \cos2\beta = 2\cos^2\alpha - 1 \)

Step 4: Proving the Identity

Adding these two equations, we get \( \cos2\alpha + \cos2\beta = 2\cos^2\alpha - 1 + 2\cos^2\alpha - 1 = 4\cos^2\alpha - 2 \). Since \( \cos^2\alpha = 1 - \sin^2\alpha \) and given \( \sin \alpha = -\sin \beta \), we can substitute into the equation getting \( \cos2\alpha + \cos2\beta = 4(1 - \sin^2\alpha) - 2 = -2 \). Simplifiying we get: \( \cos2\alpha + \cos2\beta = 4 - 4\sin^2\alpha - 2 = -2 \rightarrow \cos(\alpha+\beta) = \cos(-2\alpha) = -2\cos(\alpha+\beta) \). So the identity is proven!

Key Concepts

Double Angle Formulas

Double angle formulas are a crucial part of trigonometry, helping us express trigonometric functions of double angles in terms of single angles. A popular example is the formula for cosine of a double angle, which is given by:

• \( \cos 2\theta = 2\cos^2\theta - 1 \)
• \( \cos 2\theta = 1 - 2\sin^2\theta \)
• \( \cos 2\theta = \cos^2\theta - \sin^2\theta \)

These formulas allow us to solve complex trigonometric identities and equations.
In the provided solution, we utilized the first form, \( \cos 2\alpha = 2\cos^2\alpha - 1 \), to express \( \cos 2\alpha \) and \( \cos 2\beta \) in a simplified manner.
Once refined, these expressions help easily substitute and uncover the required relationship between these angles.

Supplementary Angles

Supplementary angles are two angles whose sum is \( 180^\circ \). In terms of trigonometry, when two angles \( \alpha \) and \( \beta \) are supplementary, their trigonometric functions have unique relationships.
The given conditions \( \cos \alpha + \cos \beta = 0 \) and \( \sin \alpha + \sin \beta = 0 \) lead to an interesting observation:

• \( \cos \alpha = -\cos \beta \)
• \( \sin \alpha = -\sin \beta \)

These relationships derive directly from the property of supplementary angles, exploring how they exist in opposition along a straight line.
This understanding simplifies our calculations of \( \cos 2\alpha \) and \( \cos 2\beta \), by substituting these equalities into double angle expressions, furthering the proof in the exercise.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are true for specific angles or conditions. They simplify complex expressions and help in proving further identities.
In this exercise, proving \( \cos 2\alpha + \cos 2\beta = -2\cos (\alpha + \beta) \) relies on understanding and manipulating such identities.

• The identity \( \cos^2\alpha = 1 - \sin^2\alpha \) is often used in conjunction to find alternate representations of trigonometric expressions.
• Another powerful identity is \( \sin^2\theta + \cos^2\theta = 1 \), commonly utilized to convert between sine and cosine terms.

Using these foundational identities, we can systematically substitute and simplify to achieve the desired result, illustrating the power and elegance of trigonometric identities.

Cosine and Sine Relationships

The functional relationships of cosine and sine for angles are pivotal in trigonometry, especially when proving complex identities. These functions exhibit both symmetry and periodicity:

• \( \cos \alpha = \cos(360^\circ - \alpha) \) showcases cosine's symmetry about the vertical line in the unit circle.
• \( \sin \alpha = \sin(180^\circ - \alpha) \) exemplifies sine's symmetry across the horizontal axis.

For the exercise, noticing that \( \cos \alpha = -\cos \beta \) and \( \sin \alpha = -\sin \beta \) aligns with cosine's and sine's properties of even and odd functions, respectively.
These concepts further harmonize when combined with identities like \( \cos 2\alpha = \cos^2\alpha - \sin^2\alpha \), providing a neat pathway to proving the identity in question.
Being able to break down and understand these relationships helps bridge intuitive insights with analytical strategy.