Problem 27 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(A+B+C=\pi\), then prove that \(\cot A+\frac{\sin A}{\sin B \sin C}=\cot B+\frac{\sin B}{\sin A \sin C}\) \(=\cot C+\frac{\sin C}{\sin A \sin B}\)

Step-by-Step Solution

Answer

The given trigonometric identity is correct and holds true for any value of A, B, and C such that A + B + C = \(\pi\).

Step 1: Initial Analysis/Transformations

The given expression to be proven has the same structure for angles A, B, and C. Thus, it is sufficient to prove it only for one of these angles. Let's consider A and express the cotangent and sine in terms of one another. Therefore, write the formulae as follows: \( \cot A = \frac{\cos A}{\sin A} \) and \( \sin A = \cos ( \pi / 2 - A) \)

Step 2: Applying Simplifications

Next, express B and C in terms of A. Because A + B + C = \(\pi\), it follows that B = \(\pi /2 - ( A + C ) ). So, we proceed to substitute B = \(\pi / 2 - ( A + C ) \) into \(\sin B), which becomes \( \sin (\pi/2 - (A + C)) \) = \( \cos (A + C) \). Similarly, \( \sin C = \cos ( \pi / 2 - C ) = \cos ( \pi /2 - ( A + B ) ) = \cos (A + B) \). The original expression for A becomes: \( \cot A + \frac{ { \cos ( \pi / 2 - A ) } }{ { \cos (A + C) \cos (A + B) } } \)

Step 3: Final Simplifications

Finally, simplify the above equation. When you substitute the identities, you should notice that it simplifies to \( \frac{ \cos A }{ \sin A } + \frac{ \sin A }{ \sin A } = \frac{ \cos A + \sin A }{ \sin A } \). After canceling out the denominator, the result is \( \cos A + \sin A \), which indeed holds true regardless of the value of A. Hence, the given equation is proved to be correct.

Key Concepts

Cotangent

Cotangent is a fundamental trigonometric function, abbreviated as "cot." It represents the reciprocal of the tangent function. Mathematically, the cotangent of an angle \( A \) is expressed as \( \cot A = \frac{\cos A}{\sin A} \). This identity illustrates the relation between the cotangent and other primary trigonometric functions: sine and cosine.

Cotangent can be particularly useful when dealing with right-angled triangles and circular functions. In the context of the exercise, cotangent helps simplify expressions by transforming them into one involving sines and cosines.

• It plays a key role when angles are part of an equation making calculations more manageable.
• By expressing cotangent in terms of sine and cosine, you can easily manipulate complex trigonometric equations.

Understanding how to work with \( \cot A \) is vital for solving more intricate problems, especially those involving identities or proofs like the one given in the exercise.

Sine Function

The sine function, commonly abbreviated as "sin," is one of the main trigonometric functions. It gives the ratio of the length of the opposite side of the angle to the hypotenuse in a right triangle. The sine function is crucial in various fields, including physics, engineering, and geometry, due to its periodic properties.

One way to express sine trigonometrically is through complementary angles. The identity \( \sin A = \cos ( \pi / 2 - A) \) shows a connection between sine and cosine, allowing transformations from one to another. In the problem, such transformations simplify the original expressions involving \( \sin B \) and \( \sin C \). This manipulation is essential:

• It provides alternative ways to express trigonometric equations.
• Enables simplification by converting sine terms into cosine terms.
• Assists in resolving complex equations by using complementary angles.

Through understanding the sine function and its properties, you can tackle problems efficiently, simplifying expressions to their basic components.

Sum of Angles

The sum of angles in a triangle is a pivotal concept in trigonometry, often leading to important identities and simplifications. A fundamental property of triangles is that the sum of interior angles is \( \pi \) (180 degrees). Thus, for angles \( A, B, \) and \( C \) in a triangle, \( A + B + C = \pi \).

This property provides the basis for many trigonometric identities and transformations. For example, knowing \( B = \pi /2 - (A + C) \) allows for simplifications that leverage the complementary angles identity. This concept lays the groundwork:

• For understanding relationships between angles within triangles.
• For deriving alternative expressions when angles complement each other.

Recognizing how angle sums influence trigonometric properties in identities, such as in the original exercise, is essential. It enables pinpointing useful simplifications and building connections between different trigonometric elements.