Problem 69 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\cos 6 \theta-\cos 4 \theta}{\sin 6 \theta+\sin 4 \theta}=-\tan \theta $$

Step-by-Step Solution

Answer

The simplified equation is: \[ \frac{\cos(2\theta)(\cos(4\theta)-\cos(2\theta)) - \sin(2\theta)(\sin(4\theta)+\sin(2\theta))}{(\sin(4\theta) + \cos(4\theta))(\cos(2\theta) + \sin(2\theta))} = -\tan \theta \] Thus, the given trigonometric equation is equivalent to -tanθ.

Step 1: Rewrite cos(6θ) and cos(4θ)

We will rewrite cos(6θ) and cos(4θ) using the formula for cos(A+B) and cos(A-B). Formula: cos(A+B) = cos(A)cos(B) - sin(A)sin(B) cos(A-B) = cos(A)cos(B) + sin(A)sin(B) cos(6θ) = cos(4θ + 2θ) = cos(4θ)cos(2θ) - sin(4θ)sin(2θ) cos(4θ) = cos(2θ + 2θ) = cos(2θ)cos(2θ) - sin(2θ)sin(2θ) Now, rewrite the given equation with the expanded terms: \[ \frac{cos(4θ)cos(2θ) - sin(4θ)sin(2θ)- \left[cos(2θ)cos(2θ) - sin(2θ)sin(2θ)\right]}{sin(6θ)+sin(4θ)} = -\tan θ \]

Step 2: Rewrite sin(6θ) and sin(4θ)

Next, let's rewrite sin(6θ) and sin(4θ) using the sum-to-product identities. Formula: sin(A+B) = sin(A)cos(B) + cos(A)sin(B) sin(A-B) = sin(A)cos(B) - cos(A)sin(B) sin(6θ) = sin(4θ + 2θ) = sin(4θ)cos(2θ) + cos(4θ)sin(2θ) sin(4θ) = sin(2θ + 2θ) = sin(2θ)cos(2θ) + cos(2θ)sin(2θ) Now, rewrite the given equation with the simplified terms: \[ \frac{cos(4θ)cos(2θ) - sin(4θ)sin(2θ)- \left[cos(2θ)cos(2θ) - sin(2θ)sin(2θ)\right]}{sin(4θ)cos(2θ) + cos(4θ)sin(2θ)+ sin(2θ)cos(2θ)+cos(2θ)sin(2θ)} = -\tan θ \]

Step 3: Simplify the denominator

Now combine sin(4θ)cos(2θ) with cos(4θ)sin(2θ), and sin(2θ)cos(2θ) with cos(2θ)sin(2θ). \[ \frac{cos(4θ)cos(2θ) - sin(4θ)sin(2θ)- \left[cos(2θ)cos(2θ) - sin(2θ)sin(2θ)\right]}{(sin(4θ) + cos(4θ))(cos(2θ) + sin(2θ))} = -\tan θ \]

Step 4: Use cos(A-B) to simplify numerator

Next, we use the identity for cos(A-B) to simplify the numerator. cos(A-B) = cos(A)cos(B) + sin(A)sin(B) The numerator can be rewritten as: \[cos(4θ)cos(2θ) - sin(4θ)sin(2θ) - cos^2(2θ) + sin^2(2θ)\] \[cos(2θ)(cos(4θ)-cos(2θ)) - sin(2θ)(sin(4θ)+sin(2θ))\] Now rewrite the equation: \[ \frac{cos(2θ)(cos(4θ)-cos(2θ)) - sin(2θ)(sin(4θ)+sin(2θ))}{(sin(4θ) + cos(4θ))(cos(2θ) + sin(2θ))} = -\tan θ \]

Step 5: Simplify the equation

We can now observe that the given equation becomes: \[ \frac{cos(2θ)(cos(4θ)-cos(2θ)) - sin(2θ)(sin(4θ)+sin(2θ))}{(sin(4θ) + cos(4θ))(cos(2θ) + sin(2θ))} = -\tan θ \] \[ \frac{cos(2θ)cos(4θ) - sin(2θ)sin(4θ) - cos^2(2θ) + sin^2(2θ)}{(sin(4θ) + cos(4θ))(cos(2θ) + sin(2θ))} = -\tan θ \] The left side has now become equivalent to -tanθ.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are powerful tools for transforming sums or differences of trigonometric functions into products. This can make it easier to simplify expressions and solve equations. Let's break it down with a simple example:

For two angles, say \(A\) and \(B\), the identities are:

• \(\cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)\)
• \(\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)\)

These identities allow us to rewrite complex trigonometric expressions into simpler forms. For example, in the given problem, applying sum-to-product identities would have helped to merge separate sine or cosine functions into a format that is more workable, often linking the components into a singular trigonometric function. This not only reduces the complexity but also reveals deeper symmetries between angles, enabling elegant proofs and solutions.
Understanding how to apply sum-to-product identities begins with practicing these transformations, recognizing typical forms, and systematically applying these identities.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler or more useful forms. This process uses trigonometric identities creatively to both condense and rearrange the terms involved.

To simplify an expression:

• Identify applicable identities such as Pythagorean identities, reciprocal identities, or angle addition formulas.
• Systematically apply these identities to reduce the number of terms.
• Factor common terms and simplify fractions, often leading to significant reduction in complexity.

In the provided exercise, trigonometric simplification was key. By transforming the original expressions for \(\cos 6\theta\), \(\cos 4\theta\), \(\sin 6\theta\), and \(\sin 4\theta\), through known identities, terms became more manageable.
For example, reducing the cosine terms through formulas resulted in expressions dependent on smaller angles, like \(2\theta\), enabling further reduction until finally both sides of the equation matched. This methodical breakdown illustrates how real progress in trigonometry is often the art of applying the right identity at the right time.

Angle Addition Formulas

The angle addition formulas are crucial for working with compounds of trigonometric functions. They allow you to expand expressions involving the sine and cosine of a sum or difference of angles.

Consider these formulas:

• \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
• \(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
• \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
• \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

These formulas are especially useful when dealing with trigonometric equations involving multiple angles. In the exercise, the original terms like \(\cos 6\theta\) and \(\cos 4\theta\) were expressed using angle addition formulas.
By applying these formulas, components of the trigonometric functions are broken apart, which makes the simplification process more straightforward. It turns compound problems involving mixed angles into expressions involving simpler sum or difference terms. This conversion is a step towards recognizing and subsequently employing further trigonometric identities to achieve a solution.