Problem 9 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that, \(\frac{3+\cot \left(76^{\circ}\right) \cot \left(16^{\circ}\right)}{\cot \left(76^{\circ}\right)+\cot \left(16^{\circ}\right)}=\cot \left(44^{\circ}\right)\)

Step-by-Step Solution

Answer

Upon simplifying the given expression, we conclude that \(\frac{3+\cot \left(76^{\circ}\right) \cot \left(16^{\circ}\right)}{\cot \left(76^{\circ}\right)+\cot \left(16^{\circ}\right)}\) indeed equals \(\cot \left(44^{\circ}\right)\).

Step 1: Convert Degrees to Radians

First, convert all angle measures from degrees to radians. \(76^{\circ} = \frac{4\pi}{15}\) radians and \(16^{\circ} = \frac{4\pi}{45}\) radians.

Step 2: Rewrite Cotangents in Terms of Sine and Cosine

The cotangent of an angle is the reciprocal of the tangent, which can also be expressed as the ratio of the cosine to the sine. This means that \(\cot (x) = \frac{\cos(x)}{\sin(x)}\). Rewrite the expression \(\frac{3+\cot \left(\frac{4\pi}{15}\right) \cot\left(\frac{4\pi}{45}\right)}{\cot \left(\frac{4\pi}{15}\right)+\cot\left(\frac{4\pi}{45}\right)}\) in this format.

Step 3: Use the Sine and Cosine Addition and Subtraction Theorems

Apply the formulas for sine and cosine of a sum and a difference to the terms in the expression. \(\cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y)\) and \(\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)\). Apply these formulas to the resulting expression from Step 2.

Step 4: Simplify the Expression

After applying the sine and cosine addition and subtraction theorems, the expression will simplify, leaving \(\cot\left(\frac{8\pi}{45}\right)\) or cotangent of \(44^{\circ}\). This is because \(\frac{8\pi}{45}\) in radians is equivalent to \(44^{\circ}\) in degrees.

Key Concepts

Cotangent Function

The cotangent function is one of the basic trigonometric functions. It is the reciprocal of the tangent function. In mathematical terms, if you have an angle \( x \), the cotangent is given by \( \cot(x) = \frac{1}{\tan(x)} \). Since tangent itself is the ratio of sine and cosine, cotangent can also be expressed as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).

This makes cotangent a very useful tool for simplifying trigonometric expressions, especially when you need to deal with ratios involving sine and cosine. Understanding cotangent helps in solving various kinds of trigonometric equations by transforming complex ratios into more manageable forms.

In practical terms, when given an equation involving cotangent, rewriting it in terms of sine and cosine can make the solution more straightforward. This manipulation often allows you to use other trigonometric identities to further simplify and solve the expression effectively.

Angle Conversion

Converting angles between degrees and radians is a fundamental skill in trigonometry. This conversion is crucial because equations and trigonometric functions often require angle measurements in radians. To convert degrees to radians, you use the formula
\( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

For instance, in the given problem, the angles \(76^{\circ}\) and \(16^{\circ}\) were converted into radians:

• \(76^{\circ} = \frac{76 \times \pi}{180} = \frac{4\pi}{15}\)
• \(16^{\circ} = \frac{16 \times \pi}{180} = \frac{4\pi}{45}\)

Converting angles is not just a mechanical task; it broadens the way you can handle a wide variety of trigonometric problems. This knowledge facilitates your use of trigonometric identities and theorems that are expressed in terms of radians.

This conversion process also aids in visualizing angles on the unit circle, where radians provide a clearer picture of angle measures and equal divisions.

Sine and Cosine Addition Theorems

The sine and cosine addition theorems are key in simplifying and proving trigonometric equations. These theorems allow you to work with sums and differences of angles efficiently. The formulas are as follows:

1. Sine of a Sum: \( \sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y) \)
2. Sine of a Difference: \( \sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y) \)3. Cosine of a Sum: \( \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \)
4. Cosine of a Difference: \( \cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y) \)

In the exercise solution, these theorems were critical in rewriting the complex trigonometric expressions in a simplified form. By substituting these identities into the expression, it becomes easier to resolve the given equation into a single value or identity, such as \( \cot(44^{\circ}) \).

Understanding how to apply these theorems is beneficial because they serve as a bridge to connect various trigonometric functions in an equation, allowing you to explore more profound relationships between angles and their respective trigonometric values. These identity tools are widely applicable in algebraic manipulation within calculus, physics, and engineering problems.