Question

\(\lim _{\mathrm{X} \rightarrow \pi}[\\{1+\operatorname{Cos}([2 \mathrm{~m}+1] \mathrm{x})\\} /\\{1+\operatorname{Cos}([2 \mathrm{n}-1] \mathrm{x})\\}]=?\) (Where \(\mathrm{m}, \mathrm{n} \in \mathrm{N}-\\{1\\})\) (a) \([(2 m+1) /(2 n-1)]^{4}\) (b) \([(2 m+1) /(2 n-1)]^{2}\) (c) \([(2 n-1) /(2 m+1)]^{2}\) (d) \([(2 n-1) /(2 m+1)]^{4}\)

Short answer

Unfortunately, none of the given answer choices match the correct answer from the step-by-step solution. The correct answer to the question is 0, which is not among the options provided. There might be a typo or error in the given choices.

Step-by-step solution

Recognize Trigonometric Properties

Since we are dealing with limits and trigonometric functions, make sure to recognize the main trigonometric properties that can be helpful in finding the limit. In this case, we will be using double-angle formulas and angle subtraction formulas.

Apply the Angle Subtraction Formula for Cosine

To simplify the expression, we can apply the angle subtraction formula for cosine to both the numerator and denominator expressions involving cosine: \[\cos(A-B) = \cos A \cos B + \sin A \sin B\] By setting \(A = [2m+1]x\) and \(B = x\) for the numerator and \(A = [2n-1]x\) and \(B = x\) for the denominator, we get: \[\lim_{x \rightarrow \pi}\frac{1 + (\cos([2m+1]x-x))}{1 + \cos([2n-1]x-x)}\]

Apply the Double-Angle Formula for Sine and Cosine

Apply the double-angle formulas for sine and cosine to the angle subtraction expressions in the numerator and denominator: \[\cos 2x = 2\cos^2 x - 1\] \[\sin 2x = 2\sin x \cos x\] Now, replace \(2x\) with \(2mx\) and \(2nx\) in the numerator and denominator, respectively. \[\lim_{x \rightarrow \pi}\frac{1 + (\cos(2mx)\cos x + \sin(2mx)\sin x)}{1 + (\cos(2nx)\cos x - \sin(2nx)\sin x)}\]

Evaluate the Limit

Evaluate the limit as x approaches π. Recall that: \[\cos(\pi) = -1 \quad \text{and} \quad \sin(\pi) = 0\] Applying these properties, we get: \[\lim_{x \rightarrow \pi}\frac{1 + (\cos(2m\pi)\cos(\pi) + \sin(2m\pi)\sin(\pi))}{1 + (\cos(2n\pi)\cos(\pi) - \sin(2n\pi)\sin(\pi))}\] \[\lim_{x \rightarrow \pi}\frac{1 - \cos(2m\pi)}{1 + \cos(2n\pi)}\] Now, notice that \(\cos(2k\pi) = 1\) for any integer k. Thus, we get: \[\lim_{x \rightarrow \pi}\frac{1 - 1}{1 + 1} = \frac{0}{2} = 0\] However, none of the provided choices equal 0. Thus, this question might have a typo or error in the given choices.

Key concepts

Trigonometric Properties

Trigonometric properties are the backbone of solving many calculus and trigonometry problems. Understanding these properties can simplify complex trigonometric expressions, making them easier to manage. These properties include fundamental identities such as

• The Pythagorean identity: \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
• The reciprocal identities: \( \sec(\theta) = \frac{1}{\cos(\theta)} \), among others.
• The co-function identities: \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \), and vice versa.

There's also the even-odd identities, which tell us that
the cosine function is even: \( \cos(-\theta) = \cos(\theta) \),
and the sine function is odd: \( \sin(-\theta) = -\sin(\theta) \).
These, among other properties, allow us to rewrite trigonometric functions in a form that can be more easily analyzed and solved, especially when approaching limits in calculus as seen in our exercise.

Angle Subtraction Formula

The angle subtraction formula is particularly useful when dealing with trigonometric functions involving the difference between two angles. It helps in transforming a complex trigonometric function into something more manageable. For cosine, the formula is given as
\[\begin{equation}\cos(A - B) = \cos A\cos B + \sin A\sin B\end{equation}\]
Applying this to limit problems allows us to break down the function into a form that usually simplifies upon taking the limit. The exercise employs this technique by setting the angles A and B in such a way that, upon applying the limit as x approaches \(\pi\), the trigonometric functions transform and become manageable in evaluating the limit.

Double-Angle Formulas

Double-angle formulas are another set of trigonometric tools that allow us to rewrite trigonometric expressions involving the double of an angle in terms of the single angle. The main double-angle formulas for sine and cosine are:

• \[\begin{equation}\cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)\end{equation}\]
• \[\begin{equation}\sin(2\theta) = 2\sin(\theta)\cos(\theta)\end{equation}\]

These formulas were applied in our step-by-step solution, transforming the trigonometric expressions and paving the way for a limit evaluation as the variable approaches \(\pi\). In the given exercise, even though the final result of the limit was 0, the process of reaching this conclusion heavily relied on utilizing the double-angle formulas to simplify the trigonometric expressions present in the numerator and denominator.