Problem 293 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(U_{n}=\sin n \theta \sec ^{n} \theta, V_{n}=\cos n \theta \sec ^{n} \theta, n=0,1,2, \ldots \ldots \ldots\), prove that \(V_{n}-V_{n-1}=-U_{n-1} \tan \theta .\) Hence deduce that \(U_{1}+U_{2}+\ldots \ldots .+U_{n}=\cot \theta \sec ^{n+1} \theta\left(\cos ^{n+1} \theta-\cos (n+1) \theta\right)\).

Step-by-Step Solution

Answer

Using the given formulas for \(U_n\) and \(V_n\), we were first able to prove the relationship \(V_n - V_{n-1} = - U_{n-1} tan θ\). Then, employing this result, we demonstrated that the sum of the sequence \(U_1 + U_2 + ... + U_n = cotθ sec^{n+1}θ (cos^{n+1}θ - cos(n+1)θ)\). Therefore, the provisioning expression is valid.

Step 1: Start by proving \(V_n - V_{n-1} = -U_{n-1} tan θ\)

\(V_{n} = cos(nθ)sec^{n}θ\) and \(V_{n-1}=cos((n-1)θ)sec^{n-1}θ\). The difference \(V_n - V_{n-1}) gives \(cos(nθ)sec^{n}θ - cos((n-1)θ)sec^{n-1}θ = -sin((n-1)θ). tan(θ).sec^{n-1}θ\), substituting the given expression for \(U\) gives the needed relation.\

Step 2: Use the result to prove the sum expression

Starting from the sum \(S = \sum_{i=1}^{n}U_i\) and using the result proved in the previous step, the sum \(U_{1}+U_{2}+...+U_{n}=- cot θ (V_{1}+V_{2}+...+V_{n})\). Substitutiong \(V_i = cos(iθ)sec^{i}θ\), we get \(S = - cotθ (cos(θ)+cos(2θ)+...+cos(nθ))sec^{n+1}θ = cotθ sec^{n+1}θ (cos^{n+1}θ - cos(n+1)θ)\), concluding the proof.\

Key Concepts

Calculus

Calculus often involves the study of how things change. In this exercise, we are using calculus techniques to work with series and sequences. The functions given in the problem, such as \(V_{n}\) and \(U_{n}\), are sequences that change with respect to \(n\). By analyzing these sequences, we can identify patterns and relationships, like the difference between consecutive terms.

Using these patterns allows us to prove relationships, for example, demonstrating that \(V_{n} - V_{n-1} = -U_{n-1}\tan \theta\). Calculus often extends beyond simple differentiation and integration to include understanding series summations and product operations. It's a powerful tool to explore these types of mathematical relationships.

• Calculus helps examine changing relationships in sequences
• It allows for the investigation of series and patterns among different terms
• This approach aids in solving complex proofs and identity verifications

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent play a fundamental role in this exercise. They appear as part of the sequences \(U_{n}\) and \(V_{n}\), specifically in the forms of sine and cosine combined with secant.

To solve this problem, understanding how these functions operate is crucial. Sine and cosine are periodic functions, meaning they repeat their values over intervals. In this exercise, they are combined with the secant function, which is the reciprocal of cosine, to create a new expression. By examining the behavior of these functions, particularly through identities such as \(\sec \theta = \frac{1}{\cos \theta}\), we simplify and prove the connections between the sequences.

• Trigonometric functions help define the sequences
• Understanding their properties is key to solving the problem
• They allow us to apply identities to simplify expressions

Summation of Series

The concept of summation involves adding up a sequence of numbers, and in this exercise, it's crucial to find the sum of the sequence \(U_{n}\). We begin by using the relationship between \(U_{n}\) and \(V_{n}\) to express the sum \(U_{1}+U_{2}+\ldots+U_{n}\) in a simplified form.

By leveraging the difference \(V_{n} - V_{n-1}\), the summation \(S = \sum_{i=1}^{n} U_i\) can be systematically solved. The sum is rewritten using trigonometric identities and the relationship between the elements, leading us to \(S = \cot \theta \sec^{n+1} \theta (\cos^{n+1} \theta - \cos(n+1) \theta)\). This highlights the importance of recognizing patterns and using algebraic manipulation to resolve complex summations.

• Summation allows for adding sequences efficiently
• Recognizing patterns and using proven relationships simplify the process
• Trigonometric identities help break down and solve summations