Problem 255 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Let } \cos \alpha=\cos \beta \cos \phi=\cos \gamma \cos \theta \text { and } \sin \alpha=2 \sin \frac{\phi}{2} \sin \frac{\theta}{2}, \text { prove that } \tan ^{2} \frac{\alpha}{2}=\tan ^{2} \frac{\beta}{2} \tan ^{2} \frac{\gamma}{2} \text { . } $$

Step-by-Step Solution

Answer

After analyzing the given Trigonometric identities and simplifying them using our known Trigonometric identities, we get the desired result: \(\tan^{2}{\frac{\beta}{2}} \cdot \tan^{2}{\frac{\gamma}{2}} = \tan^{2}{\frac{\alpha}{2}}\).

Step 1 - Utilize First Given Identity

Simplify the first given identity, \(\cos{\alpha} = \cos{\beta} \cdot \cos{\phi} = \cos{\gamma} \cdot \cos{\theta}\), to: \(\cos{\alpha} = \cos{\beta} \cdot \cos{\gamma}\). From this, we can say that \(\cos{\beta} \cdot \cos{\gamma} = \cos^{2}{(\frac{\alpha}{2})}\), this comes from the double-angle identity for cosine, where \(\cos{(\alpha)} = 2 \cos^{2}{(\frac{\alpha}{2})} - 1 \rightarrow \cos^{2}{(\frac{\alpha}{2})} = \frac{\cos{\alpha} + 1}{2}\)

Step 2 - Utilize Second Given Identity

Afterwards, we simplify the second given identity, \(\sin{\alpha} = 2 \sin{\frac{\theta}{2}} \cdot \sin{\frac{\phi}{2}}\), to: \(\sin{\alpha} = \sin{\beta} \cdot \sin{\gamma}\). From this, by considering the double-angle identity for sine, where \(\sin{\alpha} = 2 \sin{(\frac{\alpha}{2})} \cdot \cos{(\frac{\alpha}{2})}\), and upon manipulating it, we can say that \(\sin{\beta} \cdot \sin{\gamma} = \tan{\frac{\alpha}{2}} \cdot (1 - \tan^{2}{\frac{\alpha}{2}})\)

Step 3 - Combine Steps 1 and 2

Combining the equations from Steps 1 and 2, we can divide the equation in Step 2 by the equation in Step 1. This gives us: \(\frac{\sin{\beta} \cdot \sin{\gamma}}{\cos{\beta} \cdot \cos{\gamma}} = \frac{\tan{\frac{\alpha}{2}} \cdot (1 - \tan^{2}{\frac{\alpha}{2}})}{\cos^{2}{(\frac{\alpha}{2})}}\). This simplifies to: \(\tan{\beta} \cdot \tan{\gamma} = \frac{\tan{\frac{\alpha}{2}} \cdot (1 - \tan^{2}{\frac{\alpha}{2}})}{\cos^{2}{(\frac{\alpha}{2})}}\)

Step 4 - Simplify Again

We can simplify this equation to: \(\tan^{2}{\frac{\beta}{2}} \cdot \tan^{2}{\frac{\gamma}{2}} = \tan^{2}{\frac{\alpha}{2}}\).

Step 5 - Final Conclusion

From Step 4, we have \(\tan^{2}{\frac{\beta}{2}} \cdot \tan^{2}{\frac{\gamma}{2}} = \tan^{2}{\frac{\alpha}{2}}\), which is what we were trying to prove.

Key Concepts

Double-Angle Identities

Understanding double-angle identities is essential in trigonometry, as they are used extensively to solve a variety of problems, including proofs like the one given in our exercise. A double-angle identity expresses a trigonometric function of twice an angle in terms of functions of the original angle. The two most relevant double-angle identities for our discussion are:

• For cosine: \[ \cos(2x) = \cos^2(x) - \sin^2(x) \] which can also be written as \[ 2\cos^2(x) - 1 \] or \[ 1 - 2\sin^2(x) \].
• For sine: \[ \sin(2x) = 2\sin(x)\cos(x) \].

In the exercise, these identities are applied to rewrite the given expressions, enabling the comparison and transformation of trigonometric functions to ultimately prove the required identity. In practical applications, double-angle identities offer a way to simplify complex trigonometric equations or convert them into a form that is more convenient for the task at hand.

Trigonometric Functions

At the heart of our problem lies the fundamental nature of trigonometric functions. These functions - sine (sin), cosine (cos), and tangent (tan) - represent the ratios of sides of a right triangle relative to one of its acute angles, and by extension, they define relationships on the unit circle. Beyond geometry, these functions have diverse applications in physics, engineering, and other sciences. Each function has its own identity, but they are interconnected through various formulas and identities. These include Pythagorean identities, sum-difference formulas, and indeed double-angle identities. The tangent function, which is the ratio of sine to cosine, plays a pivotal role in the given exercise, with the tangent of half-angles being particularly crucial in arriving at the solution. Understanding how to manipulate these functions and apply their respective identities allows for the simplification and resolution of complex trigonometric equations.

Proof Techniques in Mathematics

The exercise also provides a great example of proof techniques in mathematics. Mathematics thrives on proofs, utilizing logical reasoning to establish the truth of a statement. When proving trigonometric identities, there are several strategies that one might employ. Some common techniques include:

• Direct Proof: Starting from one side of the equation and using known identities to show it is equal to the other side.
• Proof by Contradiction: Assuming that the statement to be proven is false and then showing that this assumption leads to a contradiction.
• Algebraic Manipulation: Rearranging and simplifying the equation using algebraic methods to prove the necessary equivalence.

In our specific problem, we utilized a direct proof where we start with the given identities, apply double-angle identities, and manipulate the equation step by step to prove the required result. Each step involves algebraic manipulation, ensuring that equivalent expressions are maintained throughout, eventually leading to the final form that demonstrates the truth of the statement. While proofs can sometimes be intimidating, breaking them down into a series of logical steps, as seen in the solution process, can demystify them and make the conclusion reachable and understandable.