Problem 16 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ (\sin A+\cos A)(\cot A+\tan A)=\sec A+\operatorname{cosec} A $$

Step-by-Step Solution

Answer

From the given trigonometric expression and using the basic trigonometric identities, we expand and simplify the expression as follows: \((\sin A+\cos A)\left(\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}\right)=\cos A+\sin A+\frac{\sin^2 A+\cos^2 A}{\cos A\sin A} = \cos A+\sin A+\frac{1}{\cos A\sin A}\) Separating fractions and rearranging terms, we get: \(\frac{1}{\cos A}+\frac{1}{\sin A}+\cos A+\sin A\) By the definitions of secant and cosecant, this expression can be written as: \(\sec A+\operatorname{cosec} A\) Thus, we have proved that: \((\sin A+\cos A)(\cot A+\tan A)=\sec A+\operatorname{cosec} A\)

Step 1: Recall basic trigonometric identities

We will be using the following trigonometric identities and relationships throughout this problem: 1. \[\cot A = \frac{1}{\tan A} = \frac{\cos A}{\sin A}\] 2. \[\tan A = \frac{\sin A}{\cos A}\] 3. \[\sec A = \frac{1}{\cos A}\] 4. \[\operatorname{cosec} A = \frac{1}{\sin A}\]

Step 2: Expand the given expression

We begin by expanding the given expression using the definitions of cotangent and tangent from Step 1: \((\sin A+\cos A)\left(\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}\right)\)

Step 3: Simplify the expression

Let's simplify the expression by distributing the sum of sine and cosine: \(\sin A\left(\frac{\cos A}{\sin A}\right)+\sin A\left(\frac{\sin A}{\cos A}\right)+\cos A\left(\frac{\cos A}{\sin A}\right)+\cos A\left(\frac{\sin A}{\cos A}\right)\) Now, cancel out some common terms for each part of the expression: \(\frac{\sin A\cos A}{\sin A}+\frac{\sin^2 A}{\cos A}+\frac{\cos^2 A}{\sin A}+\frac{\cos A\sin A}{\cos A}\) Notice that in the first term, \(\sin A\) cancels out, and in the fourth term, \(\cos A\) cancels out: \(\cos A+\frac{\sin^2 A}{\cos A}+\frac{\cos^2 A}{\sin A}+\sin A\)

Step 4: Apply trigonometric identities

Now, let's apply the trigonometric identities for secant and cosecant from Step 1: \(\frac{1}{\cos A}+\frac{1}{\sin A}=\sec A+\operatorname{cosec} A\) We need to show that our expression from Step 3 is equal to this expression. Notice that if we put the second and third terms of our expression over a common denominator, it matches the expression above: \(\cos A+\sin A+\frac{\sin^2 A+\cos^2 A}{\cos A\sin A}\) Using the Pythagorean identity, \(\sin^2 A+\cos^2 A=1\), we can simplify the expression further: \(\cos A+\sin A+\frac{1}{\cos A\sin A}\) Now, separate the fractions: \(\cos A+\sin A+\frac{1}{\cos A}+\frac{1}{\sin A}\) Rearrange the terms: \(\frac{1}{\cos A}+\frac{1}{\sin A}+\cos A+\sin A\)

Step 5: Verify equality of expressions

The expression from Step 4 is: \(\frac{1}{\cos A}+\frac{1}{\sin A}+\cos A+\sin A\) Now compare it to the given expression we were supposed to prove: \(\sec A+\operatorname{cosec} A\) By the definitions of secant and cosecant from Step 1, these two expressions are indeed equal, proving the trigonometric identity. Therefore, we have shown that: \((\sin A+\cos A)(\cot A+\tan A)=\sec A+\operatorname{cosec}A \)

Key Concepts

Trigonometric Functions

Trigonometric functions are basic concepts in trigonometry, encompassing functions like sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cotangent (\( \cot \)), secant (\( \sec \)), and cosecant (\( \operatorname{cosec} \)).
These functions are vital in understanding the relationships between the angles and sides of triangles.

• Sine, \( \sin A \), is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine, \( \cos A \), is the ratio of the adjacent side to the hypotenuse.
• Tangent, \( \tan A \), is the ratio of the opposite side to the adjacent side, which simplifies to \( \frac{\sin A}{\cos A} \).
• Cotangent, \( \cot A \), is the reciprocal of tangent: \( \frac{1}{\tan A} = \frac{\cos A}{\sin A} \).
• Secant, \( \sec A \), is the reciprocal of cosine: \( \frac{1}{\cos A} \).
• Cosecant, \( \operatorname{cosec} A \), is the reciprocal of sine: \( \frac{1}{\sin A} \).

These definitions form the building block for more complex trigonometric identities and relationships.
Understanding them allows for deeper exploration of angles and their properties in various mathematical contexts.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that links the squares of the sine and cosine functions. It is expressed as \( \sin^2 A + \cos^2 A = 1 \).
This identity is derived from the Pythagorean theorem, hence its name, and is essential for simplifying trigonometric expressions.
When working on trigonometric proofs or simplifications, the Pythagorean identity provides a crucial tool.

• It allows the replacement of \( \sin^2 A \) with \( 1 - \cos^2 A \) or vice versa, depending on the context.
• This flexibility can be particularly helpful, as seen in many problems where complex expressions are encountered.
• Through various identities similar to the Pythagorean identity, relationships between other trigonometric functions can also be derived. For example, the identity shows up in our exercise when simplifying expressions like \( \frac{\sin^2 A + \cos^2 A}{\cos A \sin A} \).

The Pythagorean identity plays a pivotal role in the calculation and simplification of trigonometric expressions.

Simplification of Expressions

Simplifying expressions in trigonometry involves using known identities to rewrite complex expressions into simpler forms. This is often necessary to reveal underlying patterns and relationships between different trigonometric terms.
In the exercise described, simplification was achieved by expanding and then combining terms into a recognizable form.

Steps to Simplification

• First, expand the given trigonometric expression by applying trigonometric identities.
• Apply cancellation where possible. For instance, reducing terms such as \( \frac{\sin A \cos A}{\sin A} \) to \( \cos A \).
• Use identities like the Pythagorean identity to further simplify the expression, as seen when replacing \( \sin^2 A + \cos^2 A \) with 1.
• Finally, reorganize terms to match a desired or given form; this was crucial in proving the provided expression \( \sec A + \operatorname{cosec} A \).

Simplification often requires creativity and a thorough understanding of the functions and identities involved. It is a skill developed through practice and is crucial for successfully tackling complex trigonometric problems.