Problem 179 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \tan \left(45^{\circ}+\frac{A}{2}\right)=\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A $$

Step-by-Step Solution

Answer

To solve the problem, it is necessary to manipulate the given equation in order to express \(\tan \frac{A}{2}\) in terms of A.

Step 1: Expand and Simplify the Left-Hand Side

Let's rewrite \(\tan \left(45^{\circ}+\frac{A}{2}\right)\) using the formula for tan of sum, which states that \(\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). So, \(\tan \left(45^{\circ}+\frac{A}{2}\right) = \frac{1 + \tan\frac{A}{2}}{1 - \tan\frac{A}{2}}\). Now, the left-hand side of the equation is \(\frac{1 + \tan\frac{A}{2}}{1 - \tan\frac{A}{2}}\).

Step 2: Simplify the Right-Hand Side

It is also advantageous to represent \(\sec A\) and \(\tan A\) in terms of \(\sin A\) and \(\cos A\). We know that \(\sec A = \frac{1}{\cos A}\) and \(\tan A= \frac{\sin A}{\cos A}\). So, the right-hand side becomes \( \frac{1 + \sin A}{\cos A}\)

Step 3: Equate and Simplify

Now we can equate the simplified right and left sides: \(\frac{1 + \tan\frac{A}{2}}{1 - \tan\frac{A}{2}} = \frac{1 + \sin A}{\cos A}\). To make this equation easier to solve, let \(\tan \frac{A}{2} = x\) which simplifies the equation to: \(\frac{1 + x}{1 - x} = \frac{1 + \sin A}{\cos A}\)

Step 4: Solve for x in terms of A

From the equation \(\frac{1 + x}{1 - x} = \frac{1 + \sin A}{\cos A}\), it is possible to isolate x ( \(\tan \frac{A}{2}\) ) and express it in terms of A. This expression is the main result of the problem

Key Concepts

Tangent Formula

The tangent formula is a fundamental trigonometric identity used to calculate the tangent of the sum of two angles. When you encounter \( \tan(A + B) \), it can be expanded using the formula:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

This identity is vital when dealing with trigonometric equations because it allows you to break down complex expressions into simpler parts. In the problem provided, it is used to simplify \( \tan \left(45^{\circ}+\frac{A}{2}\right) \) by considering \( 45^{\circ} \) and \( \frac{A}{2} \) as values for \( A \) and \( B \) respectively.
This expansion helps you translate a trigonometric expression into a form that is easier to work with, often paving the way for further simplifications or transformations in the exercise.

Secant and Tangent Relationship

The secant and tangent functions are inherently related through the identities derived from sine and cosine. To clarify this relationship, recall the definitions:

• \( \sec A = \frac{1}{\cos A} \)
• \( \tan A = \frac{\sin A}{\cos A} \)

Both expressions use \( \cos A \) as a denominator. When manipulating trigonometric identities, expressing functions like secant and tangent in terms of sine and cosine is often a powerful tool.
In the step-by-step solution, the given expression \( \sec A + \tan A \) is simplified as \( \frac{1}{\cos A} + \frac{\sin A}{\cos A} = \frac{1 + \sin A}{\cos A} \).
Such transformations are essential when comparing or equating different sides of a trigonometrical equation, allowing you to manipulate and solve it more easily by simplifying complexity into a more manageable form.

Half-Angle Identity

The half-angle identities are crucial in trigonometry for converting expressions involving angles into more workable forms. The identity for tangent of a half-angle states:

• \( \tan \frac{A}{2} = \frac{1 - \cos A}{\sin A} \)

Using these identities, complex trigonometric equations can be simplified. In the problem, the half-angle \( \frac{A}{2} \) was introduced to assist with manipulating the tangent expression into an easier format: \( \frac{1 + \tan \frac{A}{2}}{1 - \tan \frac{A}{2}} \).
By defining \( \tan \frac{A}{2} \) as \( x \), the solution further simplifies the relationship between different trigonometric expressions on either side of an equation.
Such identities enable the transformation of difficult trigonometric computations into more solvable forms by reducing angles, thus making analysis and solutions more accessible.