Problem 51 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\tan ^{-1} y=4 \tan ^{-1} x,\left(|x|<\tan \left(\frac{\pi}{8}\right)\right)\), find \(y\) as an algebraix function of \(x\) and hence prove that \(\tan \left(\frac{\pi}{8}\right)\) is a root of the equation \(x^{4}-6 x^{2}+1=0\).

Step-by-Step Solution

Answer

The algebraic function of \(x\) in terms of \(y\) is \(y = 2x\). Furthermore, \(tan(\frac{\pi}{8})\) is a root of the equation \(x^{4} - 6x^{2} + 1 = 0\).

Step 1: Solve \(tan^{-1}y = 4tan^{-1}x\) for \(y\)

To get \(y\) in terms of \(x\), we first need to transform the left side of the equation to the basic trigonometric format with the help of the tangent addition formula: \(tan(a + b) = \frac{tan(a) + tan(b)}{1 - tan(a)tan(b)}\). This formula allows us to rewrite \(4tan^{-1}x\) as \(tan^{-1}(2x) = 2tan^{-1}x\). Thus, \(tan^{-1}y = tan^{-1}(2x)\) and because the inverse tangent function has one value, it follows that \(y = 2x\).

Step 2: Substitute \(tan(\frac{\pi}{8})\) for \(x\)

Now to prove that \(tan(\frac{\pi}{8})\) is a root of the equation \(x^{4} - 6x^{2} + 1 = 0\), substitute \(x = tan(\frac{\pi}{8})\) into the equation, we get \((tan(\frac{\pi}{8}))^{4} - 6(tan(\frac{\pi}{8}))^{2} + 1\).

Step 3: Simplify the equation

We know that \(tan^{2}(A) = \frac{1 - cos(2A)}{1 + cos(2A)}\). Therefore, the equation becomes \(\frac{(1 - cos(\frac{\pi}{4}))^{2}}{(1 + cos(\frac{\pi}{4}))^{2}} - 6\frac{1 - cos(\frac{\pi}{4})}{1 + cos(\frac{\pi}{4})} + 1 = 0\). After simplification, it becomes \(0 = 0\). Therefore, \(x = tan(\frac{\pi}{8})\) is indeed a root of the equation.

Key Concepts

Tangent Addition Formula

The Tangent Addition Formula is an essential tool for solving trigonometric expressions that involve combining angles. It is particularly useful in situations where you want to express the tangent of a sum or difference of two angles using their individual tangents. The formula is given by:
\[\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\]
This formula allows us to break down complex expressions into manageable parts. For instance, in the given exercise, we used the tangent addition formula to transform the expression \(4\tan^{-1}x\).
It helped us rewrite the expression in terms of standard angles by recognizing \(\tan^{-1}(2x) = 2\tan^{-1}x\), simplifying the calculation.

• Applicable for manipulating compound angles.
• Helps in simplifying inverse trigonometric expressions.
• Used to convert multiple arcs into simpler forms.

This approach is not only mathematically elegant but also reduces potential errors in solving trigonometric problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the occurring variables. They are incredibly useful for simplifying complex trigonometric expressions and are foundational in solving trigonometric equations. Some common trigonometric identities include:

• Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Double Angle Formulas: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
• Identity for tangent squared: \(\tan^2(A) = \frac{1 - \cos(2A)}{1 + \cos(2A)}\)

In the exercise we look at, the identity \(\tan^2(A) = \frac{1 - \cos(2A)}{1 + \cos(2A)}\) is crucial.
This identity is particularly useful when simplifying the equation to prove or solve for the root. By transforming the expression \((\tan(\frac{\pi}{8}))^2\) using this identity, we ensure we handle known values like \(\cos(\frac{\pi}{4})\) efficiently. Simplifying expressions via identities is a robust strategy in trigonometry, making seemingly tough problems much more approachable.

Roots of Equations

In algebra, finding the roots of an equation involves determining the values of the variable that make the equation true. These roots can be real or complex numbers, depending on the nature of the equation.
In the context of the provided exercise, we aim to show that \(x = \tan(\frac{\pi}{8})\) is a root of the equation:
\[x^{4} - 6x^{2} + 1 = 0\]
The process involves substituting this value into the equation and verifying if the equation holds true. If substituting gives a true statement (like 0 = 0), then \(\tan(\frac{\pi}{8})\) is indeed a root.
This technique is particularly fruitful when combined with trigonometric identities, as these assist in simplifying the substitution step. It showcases how trigonometry can interweave with algebra to solve equations. Through merging these concepts, we solve complex equations with elegance and precision. Understanding equation roots helps not only in academic exercises but also in various scientific and engineering computations.