Problem 58 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin (n+1) A \sin (n-1) A+\cos (n+1) A \cos (n-1) A=\cos 2 A $$

Step-by-Step Solution

Answer

Using product-to-sum and addition formulas, we transformed the given trigonometric equation to the form: \(2 \cdot \sin (n + 1) A \sin (n - 1) A + 2 \cdot \cos (n + 1) A \cos (n - 1) A = \cos (-2A) + \cos (2A)\) After simplification and dividing both sides by 2, we get: \(\sin (n + 1) A \sin (n - 1) A + \cos (n + 1) A \cos (n - 1) A = \cos 2A\) Thus, the given equation is proved.

Step 1: Apply product-to-sum formulas

Let's rewrite the first part of the equation, \(\sin (n + 1) A \sin (n - 1) A\), using the product-to-sum formula: \[2 \cdot \sin (n + 1) A \sin (n - 1) A = \cos [(n - 1)A - (n + 1)A] - \cos [(n - 1)A + (n + 1)A]\] Now, let's rewrite the second part of the equation, \(\cos (n + 1) A \cos (n - 1) A\), using the product-to-sum formula: \[2 \cdot \cos (n + 1) A \cos (n - 1) A = \cos [(n - 1)A + (n + 1)A] + \cos [(n + 1)A - (n - 1)A]\]

Step 2: Simplify the equations

We can further simplify the two equations obtained in Step 1: For the first equation: \[2 \cdot \sin (n + 1) A \sin (n - 1) A = \cos (-2A) - \cos (2nA)\] For the second equation: \[2 \cdot \cos (n + 1) A \cos (n - 1) A = \cos (2nA) + \cos (2A)\]

Step 3: Combine the equations

Now, let's combine the two simplified equations by adding them: \[2 \cdot \sin (n + 1) A \sin (n - 1) A + 2 \cdot \cos (n + 1) A \cos (n - 1) A = \cos (-2A) - \cos (2nA) + \cos (2nA) + \cos (2A)\]

Step 4: Simplify further and divide by 2

The terms \(\cos (2nA)\) and \(- \cos (2nA)\) cancel each other out, and the equation simplifies to: \[2 \cdot \sin (n + 1) A \sin (n - 1) A + 2 \cdot \cos (n + 1) A \cos (n - 1) A = \cos (-2A) + \cos (2A)\] Now, divide both sides of the equation by 2: \[\sin (n + 1) A \sin (n - 1) A + \cos (n + 1) A \cos (n - 1) A = \frac{\cos (-2A) + \cos (2A)}{2}\] Since \(\cos (-2A) = \cos (2A)\), we have: \[\sin (n + 1) A \sin (n - 1) A + \cos (n + 1) A \cos (n - 1) A = \cos 2A\] Thus, the given equation is proved.

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are incredibly useful in trigonometry for rewriting products of sine and cosine into sums or differences of trigonometric functions. These formulas are particularly useful in simplifying expressions, solving equations, and integrating trigonometric functions.
This formula is derived from the sum and difference identities. In essence, they allow you to convert an otherwise complicated expression into something more manageable.

• For sine functions, the formula is: \( \ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \)
• For cosine functions, the formula is: \( \ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) \)

When you have a product such as \( \sin(n + 1) A \sin(n - 1) A \), applying the product-to-sum formula transforms it into a difference of two cosine terms. Similarly, a product of cosine terms can be turned into a sum of two cosine terms. Using these conversions simplifies the initial expression remarkably.

Trigonometric Simplification

Trigonometric simplification involves reducing a complex trigonometric expression to its simplest form. This process makes it easier to work with and understand the problem at hand. Simplification can involve various techniques and identities including the product-to-sum formulas, Pythagorean identities, and even symmetry properties of trigonometric functions.
In the exercise, the

• sine product was rewritten using the product-to-sum formula to \( \cos(-2A) - \cos(2nA) \)
• cosine product was converted to \( \cos(2nA) + \cos(2A) \)

The next vital step is recognizing and canceling out terms. Here, you'll notice that adding these results leads

• to cancellation of the \( \cos(2nA) \) terms
• simplifying it further to \( \cos(-2A) + \cos(2A) \)

By dividing the entire expression by 2, the solution becomes clear: the original trigonometric expression simplifies to a straightforward equation \( \cos2A \). This technique of simplifying can significantly reduce the complexity of trigonometric calculations.

Cosine Addition Formula

The cosine addition formula is a fundamental identity in trigonometry used extensively to compute the cosines of angle sums and differences. Understanding and applying this formula is crucial for solving many trigonometric problems.
The cosine addition formula states:

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

Conversely, the cosine subtraction formula indicates that:

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

These relationships are used not only in deriving product-to-sum formulas but also in numerous other trigonometric transformations.
In our exercise, while directly using the cosine addition formulas wasn't necessary, they form the foundation upon which the product-to-sum formulas stand. Respecting this hierarchical relationship helps deepen your understanding of trigonometric identities and how best to apply them in proofs or simplifications.