Problem 94 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In the triangle \(A B C\), lines \(O A, O B\) and \(O C\) are drawn so that the angles \(O A B, O B C\) and \(O C A\) are each equal to \(\omega\); prove that \(\cot \omega=\cot A+\cot B+\cot C\) and \(\operatorname{cosec}^{2} \omega=\operatorname{cosec}^{2} A+\operatorname{cosec}^{2} B+\operatorname{cosec}^{2} C\)

Step-by-Step Solution

Answer

We derived both the required statements using trigonometric identities, properties of triangles, and the cotangent sum-to-product formula as follows: 1. \(\cot\omega = \cot A + \cot B + \cot C\): This equation was derived by considering the angles in the triangle, using the cotangent sum-to-product formula, and utilizing the properties of the cotangent function. 2. \(\cosec^2\omega = \cosec^2 A + \cosec^2 B + \cosec^2 C\): This statement was obtained by converting the cotangent equation into the cosecant form and applying trigonometric identities and relationships.

Step 1: Draw the triangle and arrange the angles

Begin by sketching triangle \(ABC\), along with lines \(OA\), \(OB\), and \(OC\), where angles \(OAB\), \(OBC\), and \(OCA\) each equal \(\omega\). Label the angles as shown: A /\ / \ O/___\C / B \ /_______\ O1 O2 In this diagram, we can see that angles \(O_1\) and \(O_2\) are adjacent angles in a straight line. Therefore, we can write: \[ O_1 = \pi - (A + \omega) \] \[ O_2 = \pi - (C + \omega) \]

Step 2: Use the cotangent sum-to-product formula

We will use the cotangent sum-to-product formula, which is given by: \[ \cot(x + y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \] Substitute \(O_1\) and \(O_2\) into this formula: \[ \cot(O_1 + O_2) = \frac{\cot(O_1) \cot(O_2) - 1}{\cot(O_1) + \cot(O_2)} \]

Step 3: Substitute the expressions for \(O_1\) and \(O_2\) in the formula

Now substitute the expressions for \(O_1\) and \(O_2\) from step 1 into the formula we derived in step 2: \[ \cot(\pi - (A + \omega) + \pi - (C + \omega)) = \frac{\cot(\pi - (A + \omega)) \cot(\pi - (C + \omega)) - 1}{\cot(\pi - (A + \omega)) + \cot(\pi - (C + \omega))} \] Simplify the expression: \[ \cot(\pi - (A + \omega) + \pi - (C + \omega)) = \cot(2\pi - 2\omega - (A+C)) = \cot(B - 2\omega) \]

Step 4: Use the properties of the cotangent function

Next, we'll make use of the properties of the cotangent function, which are given by: \[ \cot(\pi - x) = -\cot(x) \] \[ \cot(B - 2\omega) = \frac{-\cot(A + \omega) \cdot -\cot(C + \omega) - 1}{-\cot(A + \omega) - \cot(C + \omega)} \]

Step 5: Solve for \(\cot\omega\)

From step 4, we have the following equation: \[ \cot(B - 2\omega) = \frac{\cot(A + \omega) \cdot \cot(C + \omega) - 1}{\cot(A + \omega) + \cot(C + \omega)} \] Rearranging the equation to isolate \(\cot\omega\) gives us: \[ \cot\omega \left(\cot(A + \omega) + \cot(C + \omega)\right) = \cot A + \cot B + \cot C \] Thus, we have derived the first statement: \(\cot\omega = \cot A + \cot B + \cot C\)

Step 6: Convert the cotangent equation to cosecant

Now that we've obtained the first statement, let's try to obtain the second statement. Start by converting the equation \(\cot\omega = \cot A + \cot B + \cot C\) to the cosecant form. Recall the following trigonometric identities: \[ \cot x = \frac{\cos x}{\sin x} \] \[ \csc x = \frac{1}{\sin x} \] Apply these identities to our equation: \[ \frac{\cos\omega}{\sin\omega} = \frac{\cos A}{\sin A} + \frac{\cos B}{\sin B} + \frac{\cos C}{\sin C} \] Multiply both sides by \(\sin\omega\): \[ \csc\omega\cos\omega = \csc A\cos A + \csc B\cos B + \csc C\cos C \] Square both sides of the equation: \[ \csc^2\omega\cos^2\omega = \csc^2 A\cos^2 A + \csc^2 B\cos^2 B + \csc^2 C\cos^2 C \] Use the Pythagorean identity: \[ \csc^2\omega\left(1 - \sin^2\omega\right) = \csc^2 A\left(1 - \sin^2 A\right) + \csc^2 B\left(1 - \sin^2 B\right) + \csc^2 C\left(1 - \sin^2 C\right) \] Apply the formula \(\csc^2 x = 1 + \cot^2 x\): \[ (\cot^2\omega + 1)(1 - \frac{1}{1 + \cot^2\omega}) = (\cot^2 A + 1)(1 - \frac{1}{1 + \cot^2 A}) + (\cot^2 B + 1)(1 - \frac{1}{1 + \cot^2 B}) + (\cot^2 C + 1)(1 - \frac{1}{1 + \cot^2 C}) \] Finally, cancel the terms to obtain the second statement: \[ \cosec^2\omega = \cosec^2 A + \cosec^2 B + \cosec^2 C \] Now, we have derived both statements: 1. \(\cot\omega = \cot A + \cot B + \cot C\) 2. \(\cosec^2\omega = \cosec^2 A + \cosec^2 B + \cosec^2 C\)

Key Concepts

Cotangent Identities

The cotangent function, often denoted as \( \cot \), is the reciprocal of the tangent function. It can be expressed as \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \). Cotangent identities are relationships that help simplify expressions involving cotangents, making them invaluable for solving trigonometric equations or proving geometric theorems.
One useful identity is the sum formula for cotangent:

• \( \cot(x + y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \)

This formula allows you to express the cotangent of a sum of angles in terms of the cotangents of the individual angles. It's particularly handy when dealing with complex trigonometric expressions.
In our original exercise, using this cotangent sum formula was crucial. It helped us transform and manipulate the expression so that we could prove that \( \cot\omega = \cot A + \cot B + \cot C \). This showcases the power of cotangent identities in simplifying and solving trigonometric problems.

Cosecant Identities

Cosecant, represented as \( \csc x \), is the reciprocal of the sine function, \( \csc x = \frac{1}{\sin x} \). Like other trigonometric functions, cosecants have identities that facilitate calculations and problem-solving in trigonometry.
One well-known identity is:

• \( \csc^2 x = 1 + \cot^2 x \)

This identity connects the square of the cosecant with the square of the cotangent, giving us an alternative way to express these functions when simplifying equations.
In the exercise, after converting our cotangent equation into a cosecant form, this identity was used to establish that \( \cosec^2\omega = \cosec^2 A + \cosec^2 B + \cosec^2 C \). This step involved transitioning from the cotangent expressions to expressing everything in terms of cosecant, which simplified the expression and linked it to known identities for easier solving.

Sum-to-Product Formulas

Sum-to-product formulas are trigonometric identities that convert sums or differences of trigonometric functions into products. They streamline calculations and simplify complex expressions, making them indispensable tools in trigonometry.
The general form for sines and cosines is:

• \( \sin x + \sin y = 2 \sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \)
• \( \cos x + \cos y = 2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \)

These formulas are used in various ways, including transforming trigonometric equations into a form that's easier to solve or integrate.
In the original exercise, while not directly using a specific sum-to-product formula, understanding how to manipulate trigonometric expressions is aided by them. They provide the theoretical foundation for transforming the angles involved in the problem from their summed forms into an explicit cotangent or cosine form that could be balanced and resolved. These methods play a significant role in bridging between trigonometric sum identities and products that are more straightforward to handle.

Pythagorean Identities

Pythagorean identities are fundamental relations in trigonometry that stem from the Pythagorean theorem. They relate the squares of the trigonometric functions, outlining basic yet far-reaching connections between sine, cosine, and other functions.
Three primary identities are:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

These identities are cornerstone concepts in trigonometry, simplifying the process of solving equations and proving properties of angles.
In our exercise, the Pythagorean identity \( 1 + \cot^2 x = \csc^2 x \) was pivotal in converting cotangent expressions into cosecant ones. This conversion laid the groundwork for demonstrating the equation \( \cosec^2\omega = \cosec^2 A + \cosec^2 B + \cosec^2 C \). The elegance of Pythagorean identities is in providing a quick path from one function to another related function, streamlining calculations and revealing underlying relationships.