Problem 9 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{1}{\cot A+\tan A}=\sin A \cos A $$

Step-by-Step Solution

Answer

The given equation is \(\frac{1}{\cot A + \tan A}\). Replace cotangent and tangent with their definitions, find a common denominator, and use the Pythagorean trigonometric identity to simplify the expression. This results in \(\sin A \cos A\), proving the given equation is true.

Step 1: Recall the definitions of cotangent and tangent

Remember that the cotangent of \(A\) is the reciprocal of the tangent of \(A\), and we know that \(\tan A = \frac{\sin A}{\cos A}\). Therefore, \(\cot A = \frac{\cos A}{\sin A}\).

Step 2: Replace cotangent and tangent with their definitions

Rewrite the left side of the equation using the definitions of cotangent and tangent: \[ \frac{1}{\frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}} \]

Step 3: Find a common denominator for the fractions in the denominator

To simplify this expression, find a common denominator for the two fractions in the denominator, which in this case will be \(\sin A \cdot \cos A\): \[ \frac{1}{\frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}} = \frac{1}{\frac{\cos^2 A + \sin^2 A}{\sin A\cos A}} \]

Step 4: Use the Pythagorean trigonometric identity

Recall the Pythagorean trigonometric identity: \(\sin^2 A + \cos^2 A = 1\). The numerator of the fraction in the denominator is the sum of the squares of \(\sin A\) and \(\cos A\), which equals 1: \[ \frac{1}{\frac{\cos^2 A + \sin^2 A}{\sin A\cos A}} = \frac{1}{\frac{1}{\sin A\cos A}} \]

Step 5: Simplify the expression

Finally, we can simplify the expression by inverting the fraction in the denominator and then multiplying the numerators and denominators: \[ \frac{1}{\frac{1}{\sin A\cos A}} = 1\cdot \sin A\cos A = \sin A \cos A \] We have now simplified the left side of the equation to match the right side of the equation. Thus, the given equation is true: \[ \frac{1}{\cot A + \tan A} = \sin A \cos A \]

Key Concepts

Understanding Cotangent

The cotangent, often abbreviated as "cot," is a fundamental trigonometric function. Cotangent is the reciprocal of the tangent function, and it has unique properties and uses that can simplify complex equations. It is defined as:

• \( \cot A = \frac{\cos A}{\sin A} \)

This definition shows that cotangent is the ratio of the cosine of angle \( A \) to the sine of angle \( A \). In simpler terms, it is the inverse of the tangent, flipping \( \sin A \) and \( \cos A \) compared to tangent. Cotangent is particularly useful when dealing with angles in right triangles or solving equations involving tangent, making it a vital part of trigonometry.
If you understand tangent and its properties, mastering cotangent becomes much easier since they are closely related.

Demystifying Tangent

The tangent function, usually denoted as "tan," is another key trigonometric function. Tangent represents the ratio of the opposite side to the adjacent side in a right-angled triangle. It is expressed in terms of sine and cosine as:

• \( \tan A = \frac{\sin A}{\cos A} \)

This definition indicates how tangent relates to the sine and cosine functions, emphasizing their interconnectedness. The tangent function is pivotal when it comes to working with slopes, angles, and in situations where both sine and cosine are involved. Understanding tangent helps to solve many trigonometric problems, such as the one in the exercise.
Tangent is especially significant in calculus for finding derivatives and in various fields of science and engineering.

Exploring the Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that stems from the Pythagorean theorem. This identity is an equation involving the square of sine and cosine, stated as:

• \( \sin^2 A + \cos^2 A = 1 \)

This identity is vital as it connects the sine and cosine functions, showing their intrinsic relationship in forming the circumference of a unit circle. It allows one to simplify and verify trigonometric expressions and equations, like in the exercise, where we used it to equate the sum of squares to one. Understanding this identity is crucial for simplifying trigonometric expressions and solving equations not only in trigonometry but also in calculus and physics.
Remembering this single identity can significantly ease your work with trigonometric problems.