Problem 277 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\text { If } \cos x=\tan y, \cos y=\tan z, \cos z=\tan x, \text { prove that } \sin x=\sin y=\sin z=2 \sin 18^{\circ}

Step-by-Step Solution

Answer

Given that cos x = tan y, cos y = tan z and cos z = tan x, the trigonometric identities and relationships can be effectively used to prove that sin x = sin y = sin z = 2 sin 18 degrees.

Step 1: Substituting cos y and cos z

Given that cos y = tan z and cos z = tan x, replace cos y and cos z in the equation cos x = tan y with the corresponding tan expressions. This results in: cos x = sin z/cos z and cos x = sin x/cos y.

Step 2: Express x, y, z in terms of sin

Rearrange above equations to find expressions for x, y and z in terms of sin, resulting in: x = arctan(sin x/cos x), y = arctan(sin y/cos y), z = arctan(sin z/cos z).

Step 3: Use sum-to-product formula for sin

We can further simplify by using the sum-to-product formula: sin(a + b) = sin a cos b + cos a sin b. The result is: sin x = sin y*cos z+cos y*sin z, sin y = sin z*cos x+cos z*sin x and sin z = sin x*cos y+cos x*sin y.

Step 4: Simplification

Notice that the three equations together can be simplified by setting all three equal to each other i.e. sin x = sin y = sin z, which gives us: sin x = 2sin y*cos z.

Step 5: Verification and final step

To verify that sin x = 2 sin 18 degrees, set y = z = 18 degrees. This gives: sin x = 2 sin 18 degrees * cos 18 degrees. Using double angle formulae, sin x = sin 36 degrees = sin (180 - 3*36) = sin 72 degrees. Since x + y + z = 180 degrees and x = y = z, it is true that x = y = z = 60 degrees, which implies sin x = sin y = sin z = sin 60 = sin (180 - 3*36) = 2 sin 18 degrees.

Key Concepts

Sum-to-Product Formula

The Sum-to-Product formula helps to transform the sum of trigonometric functions into a product. This transformation often simplifies complex expressions, making them easier to work with. For sines, the formula is:

• \( \sin a + \sin b = 2 \sin \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right) \)

In this problem, we use this formula to simplify expressions for \( \sin x, \sin y, \) and \( \sin z \). By expressing sums of angles as products, solving the trigonometric equations becomes more straightforward.
It allows us to equate \( \sin x = \sin y = \sin z \) by finding common values through simplification.

Double Angle Formula

The Double Angle Formula is crucial for transforming trigonometric functions of double angles, like \( \sin 2\theta \) and \( \cos 2\theta \). When simplifying trigonometric equations, the double angle can provide a cleaner expression. For sine, it is:

• \( \sin 2\theta = 2\sin \theta \cos \theta \)

In this exercise, utilizing the double angle formula allows us to connect \( \sin 36^{\circ} \) to \( 2\sin 18^{\circ} \cos 18^{\circ} \). This transformation aids in verifying the given equivalence of \( x = y = z = 18^{\circ} \), simplifying the relationship of the angles.

Arctan Function

The Arctan, or inverse tangent function, is used to express an angle whose tangent is a particular value. It is highly useful in trigonometry when angles need to be described in terms of sine and cosine.
In this problem, we express \( x, y, \) and \( z \) using the arctan function, such as:

• \( x = \arctan\left(\frac{\sin x}{\cos x}\right) \)
• \( y = \arctan\left(\frac{\sin y}{\cos y}\right) \)
• \( z = \arctan\left(\frac{\sin z}{\cos z}\right) \)

This expression shows how each angle relates to their corresponding sine and cosine, providing a way to compute angles directly. Understanding the arctan function ensures a comprehensive grasp of circular angle relationships.

Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions. Solving these equations often requires employing various identities and formulas to simplify and find solutions.
In this exercise, the main goal is to demonstrate that \( \sin x = \sin y = \sin z = 2\sin 18^{\circ} \). This involves setting up equations:

• \( \cos x = \tan y \)
• \( \cos y = \tan z \)
• \( \cos z = \tan x \)

By simplifying these using trigonometric identities, like the Sum-to-Product and Double Angle formulas, we solve for consistent values of \( x, y, \) and \( z \). This requires a methodical approach that emphasizes recognizing patterns and relationships within the equations.