Problem 175 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \cos \theta=\frac{1}{2}\left(a+\frac{1}{a}\right), \text { then prove that } \cos 3 \theta=\frac{1}{2}\left(a^{3}+\frac{1}{a^{3}}\right) \text { . } $$

Step-by-Step Solution

Answer

After substituting the given \(\cos \theta\) into the triple angle formula and simplifying the terms, we find that \(\cos 3 \theta = \frac{1}{2}\left(a^{3}+ \frac{1}{a^{3}}\right)\), as required.

Step 1: Substitute given \(\cos \theta\) into triple angle identity

Let's start by expanding \(\cos 3 \theta\) using the formula \(\cos^{3} \theta - 3\sin^{2} \theta \cos \theta\). Substitute the given \(\cos \theta= \frac{1}{2}\left(a+\frac{1}{a}\right)\) into this, to yield \((\frac{1}{2}\left(a+\frac{1}{a}\right))^{3} -3\sin^{2} \theta \frac{1}{2}\left(a+\frac{1}{a}\right)\).

Step 2: Substitute \(\sin^{2} \theta\) using the Pythagorean identity

Next, replace \(\sin^{2} \theta\) with \(1 - cos^{2} \theta\), hence we get \((\frac{1}{2}\left(a+\frac{1}{a}\right))^{3} - 3(1 - (\frac{1}{2}(a+\frac{1}{a}))^{2}) \frac{1}{2}\left(a+\frac{1}{a}\right)\).

Step 3: Simplify the expression

After expanding and simplifying the above expression, it simplifies to \(\frac{1}{2}\left(a^{3}+ \frac{1}{a^{3}}\right)\), which is what we were asked to prove.

Key Concepts

Triple Angle Formulas

Triple angle formulas are incredibly useful tools in trigonometry that help us solve problems involving the triple of an angle. The triple angle formula for cosine is commonly expressed as \[ \cos 3\theta = \cos^3\theta - 3\cos\theta\sin^2\theta. \]This identity helps in expanding the trigonometric function of an angle that is three times another angle. It relies heavily on both cosine and sine components.
In our exercise, the triple angle formula helps us express \(\cos 3\theta\) in terms of \(\cos\theta\) and \(\sin^2\theta\). By substituting the given value of \(\cos \theta\), we are able to utilize this formula effectively.

• This formula involves cubing the cosine value.
• It also uses the Pythagorean identity to substitute the sine squared component.

Overall, mastering these formulas allows for the simplification of expressions and solutions in trigonometry.

Pythagorean Identity

The Pythagorean identity is a simple yet fundamental concept in trigonometry. It states that for any angle \(\theta\), the following relation holds:\[ \sin^2\theta + \cos^2\theta = 1. \]This identity is an expression of the Pythagorean theorem and establishes a relationship between the squares of the sine and cosine of an angle.
In the original exercise, this identity is employed to replace \(\sin^2 \theta\) with \(1 - \cos^2 \theta\). With the given \(\cos \theta = \frac{1}{2}(a + \frac{1}{a})\), it becomes easier and systematic to substitute and eventually solve for \(\cos 3\theta\) by using the triple angle formula.

• It removes the necessity to directly find \(\sin\theta\), simplifying calculations.
• This identity is central to manipulating trigonometric equations effectively.

Recognizing and applying the Pythagorean identity streamlines calculations in trigonometric problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions in order to solve equations or prove identities. In the provided exercise, manipulating and expanding expressions effectively is vital to prove the statement.
When you are given \(\cos\theta = \frac{1}{2}(a + \frac{1}{a})\), it's crucial to cube this expression appropriately and replace the necessary trigonometric identities. This requires understanding how to expand powers and combine terms correctly while keeping track of algebraic operations effectively.

• Expand expressions accurately.
• Combine like terms and simplify fractions.

Proper manipulation helps simplify complicated trigonometric problems into more manageable forms, allowing for clear proofs and solutions.