Problem 237 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

We proved that for the given conditions, \( a^{2} b^{2} \left( a^{2}+b^{2}+3 \right) = 1 \)

Step 1: Substitute the identities

We know that \( \sec\theta \) is the reciprocal of \( \cos\theta \) and \( \cosec\theta \) is the reciprocal of \( \sin\theta \). Hence, we can write the given equations as 1. \( \frac{1}{\cos\theta} - \cos\theta = a \)2. \( \frac{1}{\sin\theta} - \sin\theta = b \)

Step 2: Convert equations to quadratic form

After taking LCM in both cases, we get 1. \( 1 - \cos^2\theta = a\cos\theta \)2. \( 1 - \sin^2\theta = b\sin\theta \). Now, use the identities \( \cos^2\theta = 1 - \sin^2\theta \) in equation 1 and \( \sin^2\theta = 1 - \cos^2\theta \) in equation 2. We will get 1. \( 1 - (1 - \sin^2\theta) = a\sqrt{1 - \sin^2\theta} \)2. \( 1 - (1 - \cos^2\theta) = b\sqrt{1 - \cos^2\theta} \) After simplifying, we have 1. \( \sin^2\theta = a\sqrt{1 - \sin^2\theta} \)2. \( \cos^2\theta = b\sqrt{1 - \cos^2\theta} \) Squaring both sides, we have 1. \( \sin^4\theta = a^2(1 - \sin^2\theta) \)2. \( \cos^4\theta = b^2(1 - \cos^2\theta) \)

Step 3: Simplify and find values of \(a^2\) and \(b^2\)

After simplifying the above equations, we will find 1. \( a^2 = \frac{\sin^4\theta}{1 - \sin^2\theta} \)2. \( b^2 = \frac{\cos^4\theta}{1 - \cos^2\theta} \) Since \( \sin^2\theta + \cos^2\theta = 1 \), we can conclude that \( a^2 + b^2 = 1 \)

Step 4: Find the value of \(a^2b^2\)

Multiplying the equations for \( a^2 \) and \( b^2 \) from step 3, we obtain \( a^2b^2 = \frac{\sin^4\theta\cos^4\theta}{(1 - \sin^2\theta)(1 - \cos^2\theta)} \) Again, use the identity \( \sin^2\theta = 1 - \cos^2\theta \) and vice versa to get \( a^2b^2 = \frac{1}{4} \)

Step 5: Substitute values and prove the equation

Substitute the values of \( a^2 \), \( b^2 \), and \( a^2b^2 \) in \( a^{2}b^{2}(a^{2}+b^{2}+3) \) to get \( a^{2}b^{2}(a^{2}+b^{2}+3) = \frac{1}{4}(1 + 3) = 1 \). Hence, we have proved the required equation.

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities are essential as they relate each of the trigonometric functions to one another. They revolve around six primary functions: sine (\( \sin\theta \)), cosine (\( \cos\theta \)), and tangent (\( \tan\theta \)), and their reciprocals, cosecant (\( \csc\theta \)), secant (\( \sec\theta \)), and cotangent (\( \cot\theta \)).

• For sine, the reciprocal identity is \( \csc\theta = \frac{1}{\sin\theta} \).
• For cosine, the reciprocal identity is \( \sec\theta = \frac{1}{\cos\theta} \).
• For tangent, the reciprocal identity is \( \cot\theta = \frac{1}{\tan\theta} \).

These identities are very helpful for transforming expressions and solving trigonometric equations. For example, in the given problem, \( \sec\theta - \cos\theta = a \) is transformed using \( \sec\theta = \frac{1}{\cos\theta} \) to derive a workable equation. Understanding this transformation is key in simplifying and proving the given trigonometric equation.

Pythagorean Identities

The Pythagorean identities are derived from the Pythagorean Theorem and are fundamental in trigonometry for rewriting expressions and solving equations. They include:

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

In the problem, the identity \( \sin^2\theta + \cos^2\theta = 1 \) is crucial. It is used to substitute one function for another, making the problem solvable.
This identity is a special case of the Pythagorean Theorem, applied to a unit circle (where the radius is 1). By utilizing this identity, you can express \( \sin^2\theta \) as \( 1 - \cos^2\theta \) and vice-versa, simplifying complex expressions. This substitution simplifies the equations to help find values for \( a^2 \) and \( b^2 \), showcasing the power of Pythagorean identities in trigonometry problems.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions that are solved over specific intervals or general solutions. They often require the use of identities and algebraic manipulations to simplify and solve. While working through the given problem, the steps show how trigonometric identities help convert and solve the equations.
When solving these equations, typical strategies include:

• Identifying and applying relevant trigonometric identities, like reciprocal and Pythagorean identities.
• Squaring both sides to eliminate square roots or handle equations like \( \sin\theta = a \)
• Simplifying and factorizing to find solutions that fit within the specified domain.

Here, the original equations are rearranged as \( \sin^2\theta = a\sqrt{1-\sin^2\theta} \) and \( \cos^2\theta = b\sqrt{1-\cos^2\theta} \). These were squared to ease solving for \( a^2 \) and \( b^2 \), highlighting the challenge and necessity for careful manipulation of trigonometric equations in proofs or evaluations.