Problem 180 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left(1+\tan \frac{\alpha}{2}-\sec \frac{\alpha}{2}\right)\left(1+\tan \frac{\alpha}{2}+\sec \frac{\alpha}{2}\right)=\sin \alpha \sec ^{2} \frac{\alpha}{2} $$

Step-by-Step Solution

Answer

After simplifying and rearranging the equation using trigonometric identities, we found out that \( \sin \alpha = 0 \)

Step 1: Factor out Each Side of the Equation

We can rewrite the equation by factoring out \( \tan \frac{\alpha}{2} \) and \( \sec \frac{\alpha}{2} \) from each side. So the equation becomes: \[ (1+\tan \frac{\alpha}{2})^2-(\sec \frac{\alpha}{2})^2 =\sin \alpha \sec ^{2} \frac{\alpha}{2} \]

Step 2: Simplify the Equation

The equation can be further simplified using the trigonometric identity \( \tan^2x+1=\sec^2x \). So we have: \[ (\sec^2 \frac{\alpha}{2})-(\sec^2 \frac{\alpha}{2}) =\sin \alpha \sec ^{2} \frac{\alpha}{2} \] which simplifies to: \[ 0=\sin \alpha \sec ^{2} \frac{\alpha}{2} \]

Step 3: Multiply both Sides by sec^2(α/2)

We can divide both sides by \( \sec ^{2} \frac{\alpha}{2} \), which gives us: \[ 0= \sin \alpha \] This is the final form of the equation.

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions that establish relationships between trigonometric functions and real numbers or other trigonometric expressions. These equations can often seem daunting, but with a strategic approach, they become much more manageable. In the given exercise, we are presented with a complex equation involving tangent and secant functions. The objective is to find the values of \( \alpha \) that satisfy the equation.

To solve such equations, one might employ a variety of techniques including factoring, using trigonometric identities, and algebraic manipulation, just as we see in the step-by-step solution provided. An important aspect when dealing with trigonometric equations is to ensure that the solution accounts for all possible angles that satisfy the equation within the given domain, which is typically the set of all angles from \( 0 \) to \( 2\pi \) when dealing with radians, or \( 0^\circ \) to \( 360^\circ \) when in degrees.

Factorization in Trigonometry

Factorization in trigonometry is a powerful tool that utilizes identities to break down complex trigonometric expressions into simpler products. This technique allows us to simplify expressions and solve equations that might otherwise be difficult to handle. The original exercise demonstrates this methodology by breaking down a composite expression into the difference of two squares, a common form of factorization.

This step is possible because of the underlying Pythagorean identity: \( \tan^2x + 1 = \sec^2x \). By recognizing the patterns and how trigonometric functions relate to one another, factorization can reduce what initially appears as a complex equation into one or more simpler equations that are easier to solve. Grasping this not only aids in problem-solving but also enhances your understanding of the interconnected nature of trigonometric functions.

Simplifying Trigonometric Expressions

The simplification of trigonometric expressions is a critical skill that requires familiarity with trigonometric identities. From the quotient and reciprocal identities to the Pythagorean identities, each plays a pivotal role in transforming and reducing expressions to their simplest forms. In our original problem, simplification is achieved by employing the Pythagorean identity, converting a trigonometric expression into an algebraic form which then cancels itself out, resulting in a much simpler equation.

Simplifying complex expressions often involves recognizing which identities are applicable and understanding that sometimes more than one identity may be necessary to fully simplify an expression. A keen eye for these identities and the relationships between trigonometric functions allows students to navigate through even the most daunting of trigonometric challenges.