Problem 82 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\sin (\theta+\phi)-2 \sin \theta+\sin (\theta-\phi)}{\cos (\theta+\phi)-2 \cos \theta+\cos (\theta-\phi)}=\tan \theta $$

Step-by-Step Solution

Answer

Applying angle sum and difference formulas and simplifying the expression, we can see that: \(\frac{\sin(\theta + \phi) - 2\sin\theta + \sin(\theta - \phi)}{\cos(\theta + \phi) - 2\cos\theta + \cos(\theta - \phi)} = \frac{-2\sin\theta}{-2\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta\) Thus, the given complex expression is equal to \(\tan\theta\).

Step 1: Use angle sum and difference formulas

We will start by applying the angle sum and difference formulas to the given expression. Angle sum and difference formulas: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) \(\sin(a - b) = \sin a \cos b - \cos a \sin b\) \(\cos(a + b) = \cos a \cos b - \sin a \sin b\) \(\cos(a - b) = \cos a \cos b + \sin a \sin b\) Applying these to our given expression: \(\frac{\sin(\theta + \phi) - 2\sin\theta + \sin(\theta - \phi)}{\cos(\theta + \phi) - 2\cos\theta + \cos(\theta - \phi)}\) \(\frac{(\sin \theta \cos \phi + \cos \theta \sin \phi) - 2\sin\theta + (\sin \theta \cos \phi - \cos \theta \sin \phi)} {(\cos \theta \cos \phi - \sin \theta \sin \phi) - 2\cos\theta + (\cos \theta \cos \phi + \sin \theta \sin \phi)}\)

Step 2: Simplifying the expression

Now, we'll continue simplifying the expression by canceling terms and factoring whenever possible: \(\frac{\sin \theta \cos \phi + \cos \theta \sin \phi - 2\sin\theta + \sin \theta \cos \phi -\cos \theta \sin \phi} {\cos \theta \cos \phi - \sin \theta \sin \phi - 2\cos\theta + \cos \theta \cos \phi + \sin \theta \sin \phi}\) \(\frac{\cancel{\sin \theta \cos \phi} + \cancel{\cos \theta \sin \phi} - 2\sin\theta + \cancel{\sin \theta \cos \phi} - \cancel{\cos \theta \sin \phi}}{\cancel{\cos \theta \cos \phi} - \cancel{\sin \theta \sin \phi} - 2\cos\theta + \cancel{\cos \theta \cos \phi} + \cancel{\sin \theta \sin \phi}}\) \(\frac{-2\sin\theta}{-2\cos\theta}\)

Step 3: Arrive at the final expression

Finally, we arrive at the given function: \(\frac{-2\sin\theta}{-2\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta\) Thus, we have demonstrated that the complex expression is equal to \(\tan\theta\).

Key Concepts

Angle Sum and Difference Formulas

The angle sum and difference formulas are crucial tools in trigonometry, allowing us to perform various trigonometric manipulations with ease. These formulas help in breaking down complex expressions by expressing trigonometric functions of sums or differences of angles in terms of products of functions of individual angles. Let's break down these essential formulas:

• For Sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
• \(\sin(a - b) = \sin a \cos b - \cos a \sin b\)
• For Cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
• \(\cos(a - b) = \cos a \cos b + \sin a \sin b\)

Using these formulas, we can expand any expression involving sums or differences of angles into simpler terms involving basic trigonometric functions.

In our problem, these expansions helped transform each part of the given expression into a form that made the simplification process possible. Breaking down the expression using these formulas is often the first step to solving more complex trigonometric equations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is essential for solving problems effectively. The goal is to reduce expressions to their simplest form, often revealing manageable patterns. This often involves canceling terms, factoring expressions, or using known identities to simplify sections of the expression.

In the original exercise, after expanding the terms using angle sum and difference formulas, the expression initially looks complicated. Noticing that many terms in the numerator and the denominator cancel out is key to simplification:

• The terms \(\sin \theta \cos \phi\) and \(\cos \theta \sin \phi\) appear twice and cancel each other out once expanded.
• This results in a much simpler form: \(-2\sin\theta\) and \(-2\cos\theta\).

Once terms have been canceled, we use the fundamental trigonometric identities to achieve the final expression.

Simplifying an expression often reduces it to a form that is easily recognizable, like the fundamental trigonometric ratios known from basic trigonometry.

Tangent Function

The tangent function plays a vital role in trigonometry. Defined as the ratio of the sine and cosine of an angle, it simplifies many expressions by converting complex trigonometric ratios into a singular term.

The function is given by:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Using the simplified expressions from our problem, where we had \(-2\sin\theta\) and \(-2\cos\theta\), we simplify this further by canceling the \(-2\) common factor:

• \(\frac{-2\sin\theta}{-2\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta\)

By showing that our complex expression simplifies to the tangent of the angle \(\theta\), we reveal the power of the tangent function in reducing and interpreting trigonometric expressions.

It signifies why understanding the basic trigonometric ratios and their properties is so important: they allow for simplified solutions to what initially appear to be complicated problems.