Problem 23 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that \(\frac{\sec ^{2} \theta-\tan \theta}{\sec ^{2} \theta+\tan \theta}\) lies between \(1 / 3\) and 3 for all real \(\theta\).

Step-by-Step Solution

Answer

Yes, it can be proven that \(\(\frac{\sec ^{2} \theta-\tan \theta}{\sec ^{2} \theta+\tan \theta}\) lies between \(1 / 3\) and 3 for all real theta.

Step 1 Express tan and sec in terms of sin and cos

Express sec^2θ and tanθ in terms of sine and cosine. Remember that \(\sec^2\theta = \frac{1}{\cos^2\theta}\) and \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). Replace sec^2θ and tanθ with these expressions in the original inequality.

Step 2 Simplify the expression

Simplifying the expression leads to \((\frac{1}{\cos^2\theta} - \frac{\sin\theta}{\cos\theta}) / (\frac{1}{\cos^2\theta} + \frac{\sin\theta}{\cos\theta})\). Now, multiply the numerator and the denominator by \(\cos^2\theta\), which would give \((1 - \sin\theta\cos\theta) / (1 + \sin\theta\cos\theta)\).

Step 3 Apply the inequality

Apply the fact that for any real x, -1 ≤ x ≤ 1. Therefore, for the term \(\sin\theta\cos\theta\), which can be any real number, it reaches its maximum when it equals 1 and its minimum when it equals -1. Substitute these values into the expression.

Step 4 Present the final answer

Substitute the maximum and minimum values. When \(\sin\theta\cos\theta = 1\), the expression becomes \((1 - 1) / (1 + 1) = 0/2 = 0\), which is less than 1/3 and 3. And, when \(\sin\theta\cos\theta = -1\), the expression becomes \((1 - (-1)) / (1 + (-1)) = 2/0\), this situation leads to division by zero which indicates infinity, greater than 3. Hence, for all other real values of \(\sin\theta\cos\theta\), the result will always lie between 1/3 and 3.

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations that hold true for all values of the variables involved. Trigonometric identities play a crucial role in simplifying complex trigonometric expressions and solving equations. Let's focus on two important identities here:

• The secant squared identity: \[ \sec^2\theta = \frac{1}{\cos^2\theta} \] This identity arises from the basic definition of the secant function, which is the reciprocal of the cosine function.
• The tangent identity: \[ \tan\theta = \frac{\sin\theta}{\cos\theta} \] This identity relates the sine and cosine of an angle to its tangent.

Using these identities, you can transform trigonometric expressions like the one in our exercise: \[ \frac{\sec ^{2} \theta-\tan \theta}{\sec ^{2} \theta+\tan \theta} \] By expressing sec and tan in terms of sine and cosine, we can further simplify the problem and derive the inequality conditions accurately.

Real Numbers

A real number includes all rational and irrational numbers, offering a comprehensive view of all possible values on the number line. These numbers can be found throughout mathematics and often represent quantities in the real world.

• Positive numbers
• Negative numbers
• Zero
• Decimals and fractions

In our exercise, real numbers are particularly significant because they represent all possible angles \(\theta\) for which the inequality must hold. An important aspect of real numbers in trigonometry is understanding how these values interact when placed within trigonometric functions like sine and cosine.
Understanding how expressions like \(\sin\theta\cos\theta\) take on different values between \(-1\) and \(1\) within the realm of real numbers is crucial when ensuring that inequalities hold.

Inequalities in Trigonometry

Inequalities are mathematical statements that show the relation between two expressions. In trigonometry, these inequalities help in defining ranges within which trigonometric expressions must lie. This is particularly useful in proving statements or solving inequalities where expressions include trigonometric functions.

• An inequality involving \(\sin\theta\cos\theta\) is considered to derive the bounds for the given expression.
• In our exercise, the inequality \(-1 \leq \sin\theta\cos\theta \leq 1\) plays a central role. This constraint arises because the maximum and minimum values the product of sine and cosine can achieve are \(1\) and \(-1\), respectively.
• For ranges like \(\frac{1}{3}\) and \(3\), understanding these bounds ensures we correctly identify intervals which the trigonometric expression respects for all real \(\theta\).

By applying these concepts of inequalities, we can effectively determine the limits and possible solution spaces for trigonometric expressions.