Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that \(\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)\) \(=\cot \left(\cot ^{-1} x+\cot ^{-1} y+\cot ^{-1} z\right)\)

Step-by-Step Solution

Answer

\(\tan(\tan^{-1}x + \tan^{-1}y + \tan^{-1}z) = \cot (\cot^{-1}x + \cot^{-1}y + \cot^{-1}z)\) can be proven by expressing tangents and cotangents in terms of sine and cosine, then applying the sum formulas and recognizing the identical form of \(\tan(a+b+c)\) and \(\cot(p+q+r)\).

Step 1: Redefine \(\tan^{-1}x, \tan^{-1}y, \tan^{-1}z\)

To simplify expressions, it's better to express them in terms of sine and cosine. Therefore, denote \(\tan^{-1}x=a\), \(\tan^{-1}y=b\), and \(\tan^{-1}z=c\). Hence, \(x=\tan a, y=\tan b, z=\tan c\).

Step 2: Evaluate \(\tan(a+b+c)\)

Apply the formula of the tangent of the sum of three angles: \(\tan(a+b+c)=\frac{\tan a + \tan b + \tan c - \tan a \tan b \tan c}{1 - \tan a \tan b - \tan b \tan c - \tan c \tan a}\). Substitute \(x=\tan a, y=\tan b, z=\tan c\), get \(\tan(a+b+c)=\frac{x+y+z-xyz}{1-x y-x z-y z}\).

Step 3: Redefine \(\cot^{-1}x, \cot^{-1}y, \cot^{-1}z\)

Denote \(\cot^{-1}x = p\), \(\cot^{-1}y = q\), \(\cot^{-1}z = r\). Hence, \(x=\cot p, y=\cot q, z=\cot r\).

Step 4: Evaluate \(\cot(p+q+r)\)

Apply the formula for the cotangent of the sum of three angles: \(\cot(p+q+r)=\frac{\cot p + \cot q + \cot r - \cot p \cot q \cot r}{1 - \cot p \cot q - \cot q \cot r - \cot r \cot p}\). Substitute \(x=\cot p, y=\cot q, z=\cot r\), get \(\cot(p+q+r)=\frac{x+y+z-xyz}{1-x y-x z-y z}\).

Step 5: Conclude

Since \(\tan(a+b+c) = \cot(p+q+r)\), hence it's proved that \(\tan(\tan^{-1}x + \tan^{-1}y +\tan^{-1}z) = \cot (\cot^{-1}x + \cot^{-1}y + \cot^{-1}z)\).

Key Concepts

Tangent and Cotangent

The tangent and cotangent functions are fundamental in trigonometry, each serving as reciprocals, yet distinctive in properties. The tangent of an angle, denoted as \(\tan\), is the ratio of the opposite side to the adjacent side in a right triangle. Meanwhile, the cotangent, expressed as \(\cot\), is the ratio of the adjacent side to the opposite side.
Some key properties include:

• \(\tan(\theta) = \frac{1}{\cot(\theta)}\)
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
• Periodicity: \(\tan(\theta + \pi) = \tan(\theta)\), \(\cot(\theta + \pi) = \cot(\theta)\)
• Range: \(\tan(\theta)\) and \(\cot(\theta)\) both vary over the entire range of real numbers.

While the tangent is undefined for angles like \(90^{\circ}\), the cotangent is undefined for angles like \(0^{\circ}\). These functions help in transforming and solving equations involving inverse trigonometric properties, making them feasible for conventional mathematical problems.

Inverse Trigonometric Functions

Inverse trigonometric functions, indicated as \(\tan^{-1}x\), \(\cot^{-1}x\), etc., are crucial for finding angles when the value of a trigonometric function is known. For instance, when we know the tangent of an angle, \(x\), \(\tan^{-1}x\) will give us the actual angle. These functions "undo" the trigonometric functions and are essential for solving equations that involve the composition of functions.
Important points to note are:

• \(\tan^{-1}(\tan(\theta)) = \theta\) if \(\theta\) is within the range \((-\frac{\pi}{2}, \frac{\pi}{2})\)
• \(\cot^{-1}(\cot(\theta)) = \theta\) if \(\theta\) is within the range \((0, \pi)\)
• The results of these functions are angles, typically expressed in radians or degrees.

These inverse functions are often used in conjunction to understand complex summations and identities, such as the one given in the original problem, ensuring correct angle calculations.

Angle Sum Formulas

Angle sum formulas are essential in trigonometry, particularly when calculating the trigonometric function of the sum of multiple angles. They enable expressions like \(\tan(a+b+c)\) to be rewritten in terms of individual tangent functions. Here's the key formula used in the exercise:For tangent:\[\tan(a+b+c) = \frac{\tan a + \tan b + \tan c - \tan a \tan b \tan c}{1 - \tan a \tan b - \tan b \tan c - \tan c \tan a}\]
Similarly, for cotangent the following formula applies:\[\cot(p+q+r) = \frac{\cot p + \cot q + \cot r - \cot p \cot q \cot r}{1 - \cot p \cot q - \cot q \cot r - \cot r \cot p}\]These formulas break down complex angle expressions into manageable parts, using known values for individual angles, which simplifies problem-solving significantly. In complex exercises like the given one, these formulas help to express and equate sums of inverse functions, making them powerful tools for proving trigonometric identities.