Problem 8 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ (\sec A+\cos A)(\sec A-\cos A)=\tan ^{2} A+\sin ^{2} A $$

Step-by-Step Solution

Answer

We cannot confirm that the given equation (\(\sec A + \cos A\))(\(\sec A - \cos A\)) = \(\tan^2 A + \sin^2 A\) holds true. After simplification, the left side of the equation is actually equal to \(\sin^2 A = \tan^2 A \cdot \cos^2 A\), which is not the same as the right side of the original equation.

Step 1: Write down the given problem

We need to verify whether the equation: (\(\sec A + \cos A\))(\(\sec A - \cos A\)) = \(\tan^2 A + \sin^2 A\)

Step 2: Simplify the left side of the equation

To do this, we need to expand the left side of the equation using the identity, (a+b)(a-b) = \(a^2 - b^2\). So, we have: \( (\sec A + \cos A)(\sec A - \cos A) = \sec^2 A - \cos^2 A \)

Step 3: Convert secant to cosine and tangent

Recall the definition of secant and tangent functions: \(\sec A = \frac{1}{\cos A}\) \(\tan A = \frac{\sin A}{\cos A}\) With these in mind, we can rewrite the left side of the equation. \( \sec^2 A - \cos^2 A = (\frac{1}{\cos^2 A}) - \cos^2 A \)

Step 4: Simplify the expression

In order to compare this expression to the right side of the equation, we need a common fraction denominator: \( \frac{1}{\cos^2 A} - \cos^2 A = \frac{1 - \cos^4 A}{\cos^2 A} \)

Step 5: Use trigonometric identity

We can use the Pythagorean identity: \( \sin^2 A + \cos^2 A = 1 \), which means \( \sin^2 A = 1 - \cos^2 A \), to transform our expression. Let's substitute this into our expression: \( \frac{1 - \cos^4 A}{\cos^2 A} = \frac{(\cos^2 A)(1 - \cos^2 A)}{\cos^2 A} = \frac{(\cos^2 A)(\sin^2 A)}{\cos^2 A} \)

Step 6: Simplify further

Now, we can cancel out \(\cos^2 A\) from the numerator and the denominator: \( \frac{(\cos^2 A)(\sin^2 A)}{\cos^2 A} = \sin^2 A \)

Step 7: Convert sine to cosine and tangent

Recall the definition of tangent function again: \(\tan A = \frac{\sin A}{\cos A}\) We can change the expression to: \(\sin^2 A = \tan^2 A \cdot \cos^2 A\)

Step 8: Compare with the right side of the equation

After simplifying the left side of the equation, we obtained: \( \sin^2 A = \tan^2 A \cdot \cos^2 A \) Now compare it to the right side of the original equation: \(\tan^2 A + \sin^2 A\) Unfortunately, these two expressions are not equal. Therefore, we cannot confirm that the given equation holds true.

Key Concepts

Secant and Cosine Identity

The secant and cosine identity is a fundamental relationship in trigonometry where the secant of an angle is the reciprocal of the cosine of that angle. This means

• \( \sec A = \frac{1}{\cos A} \)
• Similarly, \( \cos A = \frac{1}{\sec A} \)

This reciprocal relationship highlights how secant and cosine are connected as inverse trigonometric functions.
You often see secant used in identities to simplify and solve equations since it transforms cosine-based expressions into reciprocal terms for easier manipulation. Recognizing this relationship is crucial in trigonometric simplifications.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry. It states that for any angle \( A \), the sum of the squares of sine and cosine will always equal one:

• \( \sin^2 A + \cos^2 A = 1 \)

This identity is particularly useful for simplifying expressions or solving equations involving squared trigonometric functions.
In our problem, we use \( \sin^2 A = 1 - \cos^2 A \), which is a rearrangement of the Pythagorean identity. Recognizing and applying this identity can help reduce complex trigonometric expressions into simpler forms, making them easier to compare or equate.

Tangent Function

The tangent function expresses the ratio of the sine to the cosine of an angle:

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

Tangent is especially useful when dealing with right triangles where you're interested in the steepness or slope, as it compares the opposite side to the adjacent side.
In the exercise presented, it is integral when expressing the relationship between sine squared and tangent squared terms. By manipulating these terms, and using the knowledge that \( \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \), we can transform expressions to help verify identities or prove equalities.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions using identities and algebraic manipulation to make equations easier to handle or compare. It often includes steps like

• Expanding products using formulas like \((a+b)(a-b) = a^2 - b^2\)
• Substituting identities such as the secant-cosine relationship and Pythagorean identities
• Canceling common terms

In this specific problem, simplification started by expanding \((\sec A + \cos A)(\sec A - \cos A)\) using the difference of squares method. Subsequently, we used various identities to transform and reduce expressions. Mastery of these techniques is essential as they allow you to manage and solve trigonometric equations more efficiently.