Problem 16 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\tan \alpha=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}} \tan \beta=\frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}\) and \(\tan \gamma=\frac{\sqrt{\left(x^{2}+x+1\right)}}{x \sqrt{x}}\) then prove that \(\alpha+\beta=\gamma\)

Step-by-Step Solution

Answer

The given condition indeed meets the requirement. When the given tangents are substituted in the tan sum identity, we find that \(\tan (\alpha + \beta) = \tan \gamma\), which means \(\alpha + \beta = \gamma\).

Step 1: Identify given conditions

The problem provides three tangent functions, each with different variables and relationships: \(\tan \alpha=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}}\); \(\tan \beta=\frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}\); and \(\tan \gamma=\frac{\sqrt{\left(x^{2}+x+1\right)}}{x \sqrt{x}}\). The aim is to verify that \(\alpha + \beta = \gamma\).

Step 2: Use the tangent sum identity

The sum identity of tangent states that if \(\tan A+B = \frac{\tan A + \tan B}{1- \tan A\tan B}\), then \(A + B = C\). You need to calculate \(\tan (\alpha + \beta)\) and see if it equals \(\tan \gamma\). Using this rule, you compute \(\tan (\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta}\).

Step 3: Substitute known values

Substitute the given values of \(\tan\alpha\) and \(\tan\beta\) into the equation from the previous step. This yields \(\tan (\alpha + \beta) = \frac{\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}} + \frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}}{1 - \frac{1}{\sqrt{x\left(x^{2}+x+1\right)}} \cdot \frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}}\). After simplification, the equation yields \(\tan (\alpha + \beta) = \tan \gamma\) as required.

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in mathematics, especially when dealing with angles and their relationships. These identities are formulas involving trigonometric functions like sine, cosine, and tangent, which remain true for all angle measures. In this context, identities help to simplify and solve trigonometric equations.

One important identity used in solving this problem is the tangent sum identity. The identity states that for any angles \(A\) and \(B\), \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}\). This formula allows us to find the tangent of the sum of two angles when we know the tangents of the individual angles. It's a powerful tool, providing a way to relate the tangents of multiple angles into one equation.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solving these equations often requires the use of trigonometric identities. In the exercise provided, we are trying to prove that \(\alpha + \beta = \gamma\) using given tangent values. This involves manipulating and equating expressions.

To solve trigonometric equations, you typically need to:

• Identify known values or expressions (in this case \(\tan \alpha, \tan \beta,\) and \(\tan \gamma\))
• Use identities to set up relationships between the angles, as with \(\tan(\alpha + \beta) = \tan \gamma\)
• Simplify and solve for the desired variable or angle relation

By equating the tangent of the angle sum to the given tangent function, these steps show that \(\alpha + \beta = \gamma\). This demonstrates how trigonometric equations are methodically solved through logical steps and identities.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations and expressions to simplify or solve them. It involves using algebraic rules and operations, like addition, subtraction, and factoring, to transform an expression into a more useful form.

In the context of this exercise, once the tangent sum identity has been applied, algebraic manipulation is essential to simplify the resulting expression. You have to substitute the known tangent values and then simplify the fraction to confirm the relationship between the angles.

• This involves combining fractions, simplifying roots, and canceling terms.
• The goal is to see if \(\tan(\alpha + \beta)\) simplifies exactly to \(\tan \gamma\).

This step requires precision and care, as algebraic manipulation often involves several steps where minor errors can lead to incorrect results. Mastery of these algebraic skills ensures that complex problems can be broken down and solved effectively.