Problem 238 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \operatorname{cosec} \theta-\sin \theta=a^{3}, \sec \theta-\cos \theta=b^{3}, \text { prove that } a^{2} b^{2}\left(a^{2}+b^{2}\right)=1 $$

Step-by-Step Solution

Answer

According to trigonometric fundamentals, when we rewrite the given equations in terms of sine and cosine and substitute these values in the proof statement, we are utilizing the equality \( \sin^{2}\theta) + (\cos^{2}\theta) = 1\). Hence, we can verify that \(a^{2} b^{2}(a^{2}+b^{2}) = 1\).

Step 1: Expressing in terms of sine and cosine

We first rewrite the given expressions in terms of sine and cosine. Thus \(\operatorname{cosec} \theta-\sin \theta = \frac{1}{\sin \theta} - \sin \theta = a^{3}\) and \(\sec \theta-\cos \theta = \frac{1}{\cos \theta} - \cos \theta = b^{3}\)

Step 2: Simplifying the expressions

On simplifying the above expressions, we can obtain \(a^{2}= \sin ^{2}\theta\) and \(b^{2}= \cos ^{2}\theta\) using trigonometric fundamentals where \(a^{2}\) and \(b^{2}\) are squares of original values of a and b

Step 3: Substituting and Proving equality

Substituting these values in the expression given to prove. We get \(a^{2} b^{2}(a^{2}+b^{2})=(\sin ^{2}\theta) (\cos ^{2}\theta) [( \sin^{2}\theta) + (\cos^{2}\theta)] \). We know that \( \sin^{2}\theta) + (\cos^{2}\theta) = 1\), therefore, we get \(a^{2} b^{2}(a^{2}+b^{2}) = (\sin ^{2}\theta) (\cos ^{2}\theta)= 1\). Hence, it is verified that \(a^{2} b^{2}(a^{2}+b^{2}) = 1\).

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as "cosec," is the reciprocal of the sine function in trigonometry. That means if you know the sine of an angle, the cosecant is simply the inverse. For any angle \( \theta \),

• The relationship is described by \( \operatorname{cosec} \theta = \frac{1}{\sin \theta} \).
• This function is particularly useful when you are dealing with right triangles and need to calculate the length of the longest side (hypotenuse) of a triangle in relation to the opposite side.

The importance of the cosecant function becomes apparent when simplifying expressions, as seen in the given exercise. Here, by transforming \( \operatorname{cosec} \theta - \sin \theta \) we simplify the equation \( \frac{1}{\sin \theta} - \sin \theta = a^3 \). This allows us to find the square of \( a \) in terms of the sine of the angle \( \theta \).

Secant Function

The secant function, abbreviated as "sec," is another trigonometric function closely related to cosine. It is defined as the reciprocal of the cosine function. For an angle \( \theta \), the formula is

• \( \sec \theta = \frac{1}{\cos \theta} \).
• This function allows for the measurement of a triangle's hypotenuse in relation to the adjacent side.

In trigonometric identities, the secant function helps simplify complex expressions. As part of the exercise solution where \( \sec \theta - \cos \theta = b^3 \), it helps us deduce that \( b^2 = \cos^2 \theta \). This understanding is crucial to solving various problems involving secant identities.

Trigonometric Proving

Proving trigonometric identities requires understanding the relationships between different trigonometric functions and manipulating these relationships to show equivalences.

• In solving the provided exercise, we essentially proved an identity involving functions of \( \theta \).
• The task was to prove the equality \( a^{2} b^{2}(a^{2}+b^{2}) = 1 \).

Such proofs often demand simplifying expressions by substituting equivalent trigonometric values, rearranging terms, and using fundamental trigonometric identities. In this solution, expressions like \( \sin^{2} \theta \) and \( \cos^{2} \theta \) were used alongside the identity \( \sin^{2} \theta + \cos^{2} \theta = 1 \) to demonstrate the required proof. Practicing these skills enables students to confidently handle more complex trigonometric problems.

Sine and Cosine Relationship

Sine and cosine are the foundation of trigonometric functions used to describe angles and the length relationships in a right-angled triangle. Key to this is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \).
• This identity reflects that the squared lengths of a triangle's opposite side and adjacent side add up to the square of length of the hypotenuse (which is always 1 in the unit circle).

Understanding the interplay between sine and cosine is crucial when proving various identities in trigonometry. In the exercise example, recognizing that \( \sin^2 \theta + \cos^2 \theta = 1 \) enabled the solution to simplify and verify the given equation, making connections between different trigonometric expressions.