Problem 39 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In any triangle \(A B C\), if \(\cos \theta=\frac{a}{b+c}\), \(\cos \varphi=\frac{b}{a+c}, \cos \psi=\frac{c}{a+b}\), where \(\theta, \varphi \& \psi\) lie between 0 and \(\pi\), prove that, \(\tan ^{2}\left(\frac{\theta}{2}\right)+\tan ^{2}\left(\frac{\varphi}{2}\right)+\tan ^{2}\left(\frac{\psi}{2}\right)=1\)

Step-by-Step Solution

Answer

\( \tan^2 \frac{\theta}{2} + \tan^2 \frac{\varphi}{2} + \tan^2 \frac{\psi}{2}= 1\)

Step 1: Express Cosines in terms of Tangents

Express each cosine equation using double-angle formula: \( \cos \theta=1-2 \sin^2 \frac{\theta}{2} \). We then have: \( a = (b+c)(1-2 \sin^2 \frac{\theta}{2}) \), \( b = (a+c)(1-2 \sin^2 \frac{\varphi}{2}) \), \( c = (a+b)(1-2 \sin^2 \frac{\psi}{2}) \)

Step 2: Express Sines in terms of Tangents

Now, we can express each sine in terms of tangents using the identity: \( \sin x = \frac{tan x}{\sqrt{1+ tan^2x}} \), so we have: \( a = (b+c)(1-2 \frac{\tan^2 \frac{\theta}{2}}{1 + \tan^2 \frac{\theta}{2}}) \), \( b = (a+c)(1-2 \frac{\tan^2 \frac{\varphi}{2}}{1 + \tan^2 \frac{\varphi}{2}}) \), \( c = (a+b)(1-2 \frac{\tan^2 \frac{\psi}{2}}{1 + \tan^2 \frac{\psi}{2}}) \)

Step 3: Simplify the equations

We simplify each equation and move to one side: $a(1+\tan^2 \frac{\theta}{2}) - 2b\tan^2 \frac{\theta}{2} - 2c\tan^2 \frac{\theta}{2} = 0$, $b(1+\tan^2 \frac{\varphi}{2}) - 2a\tan^2 \frac{\varphi}{2} - 2c\tan^2 \frac{\varphi}{2} = 0$, $c(1+\tan^2 \frac{\psi}{2}) - 2a\tan^2 \frac{\psi}{2} - 2b\tan^2 \frac{\psi}{2} = 0$.

Step 4: Add the equations

Adding the three equations gives us: $a(1+\tan^2 \frac{\theta}{2}) + b(1+\tan^2 \frac{\varphi}{2}) + c(1+\tan^2 \frac{\psi}{2}) = 2(a+b+c)\left(\tan^2 \frac{\theta}{2} + \tan^2 \frac{\varphi}{2} + \tan^2 \frac{\psi}{2}\right)$.

Step 5: Rearrange the obtained equation

We know that \(a+b+c\) is always greater than 0, so we could cancel them from both sides, which gives us the desired identity: \( \tan^2 \frac{\theta}{2} + \tan^2 \frac{\varphi}{2} + \tan^2 \frac{\psi}{2}= 1\).

Key Concepts

Triangle identities

In trigonometry, understanding triangle identities helps solve problems efficiently by relating the sides and angles of a triangle. For example, we can use trigonometric functions such as sine, cosine, and tangent to describe the relationships between angles and sides.
Triangle identities are particularly powerful when working with both right and non-right triangles, allowing mathematical expressions to be simplified or re-structured.
Commonly used triangle identities include the Pythagorean identity and the angle sum or difference identities. These identities help in connecting various trigonometric functions and establishing equivalence in trigonometric expressions.

Double-angle formula

The double-angle formula is a crucial trigonometric identity that allows us to express trigonometric functions of twice an angle in terms of trigonometric functions of the original angle.
For instance, the double-angle formula for cosine is given by:

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• Alternatively: \( \cos 2\theta = 2\cos^2 \theta - 1 \)

These formulas are extremely useful in transforming and simplifying expressions that involve trigonometric functions. Double-angle formulas are implementational when solving trigonometric equations, integrating, or proving identities through intermediate transformations.
Knowing these formulas well supports a deeper understanding of trigonometric relationships and can be critical when solving complex problems.

Trigonometric identities

Trigonometric identities are mathematical equations that involve trigonometric functions, which hold true for all values of the involved variables within their domains.
These identities, such as the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \), are foundational to understanding more complex trigonometric problems.
There are several types of identities:

• Pythagorean identities
• Angle sum and difference identities
• Double and half-angle identities
• Product-to-sum and sum-to-product identities

These identities simplify the calculation and transformation of trigonometric expressions, making advanced problems more approachable. They form the backbone for developing further theorems and solving equations in trigonometric contexts.

Sine and cosine relationships

Sine and cosine are fundamental trigonometric functions that describe the ratios of sides in right-angled triangles.
These functions are also periodic and have a unique and important relationship with each other, often described through identities such as:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( \sin(\theta + \phi) = \sin\theta \cos\phi + \cos\theta \sin\phi \)
• \( \cos(\theta + \phi) = \cos\theta \cos\phi - \sin\theta \sin\phi \)

Sine and cosine can also be used to express other trigonometric functions, like tangent, where \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).
These relationships and identities are key to understanding and solving both theoretical and applied trigonometric problems. They aid in simplifying expressions and are instrumental in proving other deeper mathematical results.