Problem 29 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

The transformed expression is \( \frac{1 + \sin A}{\cos A} \), and it is indeed equal to the initial expression \( \frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} \).

Step 1 - Rewrite in terms of sine and cosine

Substitute \( \tan A = \frac{\sin A}{\cos A} \) and \( \sec A = \frac{1}{\cos A} \) in the given equation. Then we get \( \frac{\frac{\sin A}{\cos A}+\frac{1}{\cos A}-1}{\frac{\sin A}{\cos A}-\frac{1}{\cos A}+1}=\frac{1+\sin A}{\cos A} \)

Step 2 - Simplify the Fractions

Doing so, the left side becomes \( \frac{\sin A + 1 - \cos A}{\sin A - 1 + \cos A} = \frac{1 + \sin A}{\cos A} \). Simplify the fraction in the left side by multiplying top and bottom by \( \cos A \), we get \( \frac{1 + \sin A}{\cos A} = \frac{1 + \sin A}{\cos A} \)

Step 3 - Verify

At this point, both sides of the equation are the same, so our transformations are complete and have been verified. We can say that \( \frac{\tan A+\sec A-1}{\tan A-\sec A+1} = \frac{1+\sin A}{\cos A} \).

Key Concepts

Sine and Cosine

Understanding sine and cosine is essential for solving trigonometric equations. Sine and cosine are fundamental trigonometric functions that relate to angles in a right triangle. They connect the angle with the ratios of two sides of the triangle.
\( \sin A \) represents the ratio of the opposite side to the hypotenuse, while \( \cos A \) is the ratio of the adjacent side to the hypotenuse. These functions are crucial when we express other trigonometric functions like tangent and secant.
In this exercise, we express \( \tan A = \frac{\sin A}{\cos A} \) and \( \sec A = \frac{1}{\cos A} \) using sine and cosine. This allows us to manipulate the equation more easily and ensure we are working with compatible expressions throughout the problem.

Tangent and Secant

Tangent and secant are trigonometric functions that derive from sine and cosine. Tangent, noted as \( \tan A \), is defined as the ratio of sine to cosine, \( \tan A = \frac{\sin A}{\cos A} \). In right triangles, this represents the ratio of the opposite side to the adjacent side.
Secant, written as \( \sec A \), is the reciprocal of cosine: \( \sec A = \frac{1}{\cos A} \).

By expressing these functions in terms of sine and cosine, we simplify the manipulation of trigonometric expressions. In the original exercise, substituting these definitions simplifies the process of solving and verifying the equation. Understanding these relationships is a key step in mastering trigonometric identities.

Fraction Simplification

Simplifying fractions is a valuable technique in algebra and trigonometry. When working with trigonometric identities, fractional simplification is crucial. The given problem involves simplifying the fraction \( \frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} \) to become \( \frac{1 + \sin A}{\cos A} \).

By rewriting tangent and secant in terms of sine and cosine, we can express the fraction with the common denominator \( \cos A \).

• This involves substituting \( \tan A = \frac{\sin A}{\cos A} \) and \( \sec A = \frac{1}{\cos A} \).
• After substitution, combine terms and multiply by \( \cos A \) to simplify the numerator and denominator separately.

This process ultimately allows both sides of the equation to match, indicating that they are equivalent.

Equation Verification

Equation verification involves confirming that both sides of an equation are equivalent. This technique is essential in verifying trigonometric identities. In the exercise at hand, verifying the equation \( \frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \frac{1 + \sin A}{\cos A} \) required simplifying and rewriting in terms of sine and cosine.

After substituting the trigonometric identities and simplifying the fractions, both sides of the equation simplify to \( \frac{1 + \sin A}{\cos A} \). This shows that each transformation applied to the original equation preserves its equality.
Whenever you verify an equation, ensure that each step logically follows and that no part of the expression alters the balance between both sides, unless it's intended for simplification. Equation verification confirms the integrity of our transformations and provides confidence in our solutions.