Problem 263 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \cos \theta+\sin \theta=\sqrt{2} \cos \theta, \text { show that } \cos \theta-\sin \theta=\sqrt{2} \sin \theta $$

Step-by-Step Solution

Answer

\(\cos \theta-\sin \theta = \sqrt{2} \sin \theta\).

Step 1: Re-arrange the equation

Rearrange the original equation \(\cos \theta+\sin \theta=\sqrt{2} \cos \theta\) to get \( \sin \theta=\sqrt{2}\cos \theta - \cos \theta\). This can be done by subtracting \(\cos \theta\) from both sides.

Step 2: Simplify the equation

Factor out \(\cos \theta\), you will get \(\sin \theta = \cos \theta (\sqrt{2}-1)\). The right side is now much simpler.

Step 3: Use Pythagorean identity

Now you can simplify the equation \(\cos \theta-\sin \theta=?\) by replacing \(\sin \theta\) with the previously obtained equation, which gives us \( \cos \theta - \cos \theta (\sqrt{2}-1)=?\).

Step 4: Simplify the expression

Now simplify \(\cos \theta - \cos \theta (\sqrt{2}-1)\) to \(\cos \theta \times 1-\cos \theta \times (\sqrt{2} - 1) =\cos \theta \times (1-\sqrt{2} + 1)\) = \(2 \cos \theta - \sqrt{2} \cos \theta\).

Step 5: Factor out

Factor out to get \( \cos \theta \times (2 -\sqrt{2})\).

Step 6: Recalling the trigonometric identity

Recalling the value obtained in Step 2 i.e., \( \sin \theta=\sqrt{2}\cos \theta - \cos \theta\), we can see that \( \cos \theta \times (2 -\sqrt{2})\) equals \( \sqrt{2} \sin \theta\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, crucial for understanding many trigonometric equations and expressions. It states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) is always true.
This identity derives from the Pythagorean Theorem and serves as a foundational building block in simplifying trigonometric expressions.

To apply the Pythagorean Identity, remember to:

• Recognize when an expression contains \( \sin^2 \theta \) or \( \cos^2 \theta \) and replace it using the identity
• Rearrange the expression to simplify or solve more complex equations
• Use it to transform equations to help isolate terms or factor expressions

Understanding this identity allows for a smoother transition to solving equations and simplifying more complex trigonometric expressions.

Trigonometric Equations

Trigonometric Equations involve unknown angles that satisfy a condition based on trigonometric functions like sine, cosine, or tangent. In the given exercise, we started with the equation \( \cos \theta + \sin \theta = \sqrt{2} \cos \theta \), which involves an angle \( \theta \) and its trigonometric functions.

These equations often need creative rearrangement or factoring to solve. Here are some steps to tackle trigonometric equations effectively:

• Identify the terms and organize them to isolate a trigonometric function
• Simplify the equation by factoring or using identities such as the Pythagorean Identity
• If necessary, express a function in terms of another and solve for the variable

These steps were used in solving the original exercise. Factorization and recognizing common variables allowed us to transform the equation to reveal a true statement, thus solving for \( \theta \).

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a valuable skill for solving equations efficiently. The aim is to break down complex trigonometric terms into simpler components.
This often involves using identities, factoring, and rearranging terms, as seen in the solution steps.

To simplify a trigonometric expression effectively, consider:

• Using trigonometric identities like the Pythagorean Identity to replace or eliminate complex terms
• Factoring out common terms to reduce the number of variables
• Rewriting complex expressions in simpler equivalent forms

In the exercise, rearrangement and factorization transformed a complex equation into an equivalent one, demonstrating the power of simplification in making equations solvable. Recognizing when and how to simplify helps streamline solving trigonometric problems, making them less intimidating.